ฟังก์ชันตรีโกณมิติ

ชื่อ-สกุล………………………………………..……………………… ชัน้ มัธยมศึกษาปีที่ 5/……. เลขที…่……..

หน่วยการเรียนรู้ที่ 1 เรื่องฟังก์ชันตรีโกณมิติ กลุ่มสาระการเรียนรู้คณิตศาสตร์ หน้า 1

1. ก าหนดรูปสามเหลี่ยมมุมฉากดังรูป จงบอกชื่อด้านตามชื่อที่ก าหนด 2.

รูปสามเหลี่ยม ชื่อของด้าน A B C

ด้านตรงข้ามมุมฉาก คือ ด้าน …………………. ด้านตรงข้ามมุม A คือ ด้าน …………………. ด้านประชิดมุม A คือ ด้าน …………………. ด้านตรงข้ามมุม C คือ ด้าน …………………. ด้านประชิดมุม C คือ ด้าน ………………….

A C B

ด้านตรงข้ามมุมฉาก คือ ด้าน …………………. ด้านตรงข้ามมุม A คือ ด้าน …………………. ด้านประชิดมุม A คือ ด้าน …………………. ด้านตรงข้ามมุม C คือ ด้าน …………………. ด้านประชิดมุม C คือ ด้าน ………………….

N M O

ด้านตรงข้ามมุมฉาก คือ ด้าน …………………. ด้านตรงข้ามมุม O คือ ด้าน …………………. ด้านประชิดมุม O คือ ด้าน …………………. ด้านตรงข้ามมุม M คือ ด้าน …………………. ด้านประชิดมุม M คือ ด้าน ………………….

Q P R

ด้าน PQ คือ ………………………………….. ด้าน PR คือ ………………………………….. ด้าน RQ คือ ………………………………….. ด้าน PR คือ ………………………………….. ด้าน RQ คือ …………………………………..

Y Z X

ด้าน XZ คือ ………………………………… ด้าน YZ คือ …………………………มุม Z ด้าน XY คือ …………………………มุม Z ด้าน XY คือ …………………………มุม X ด้าน YZ คือ ………………………….มุม X

จากอัตราส่วนตรีโกณของรูปสามเหลี่ยมที่ก าหนดให้ จงเติมค าตอบให้สมบูรณ์

ทบทวนความรู้เดิม เรื่อง อัตราส่วนตรีโกณมิติ



รูปสามเหลี่ยม อัตราส่วนของความยาวด้าน

A B C

มฉากนตรงขา้มมุความยาวด้ามCนตรงขา้มมุความยาวด้า

ACAB

= …………….

Ccosมฉากนตรงขา้มมุความยาวด้า

Cนประชดิมุมความยาวด้าACBC

CtanCนประชดิมุมความยาวดา้มCนตรงขา้มมุความยาวดา้

BCAB

A C B

BC

AB ………………………….. ,

BC

AC …………………………..

AC

AB………………………….. ,

BC

AC …………………………..

BC

AB ………………………….. ,

AB

AC…………………………..

N M O

O.........MOMN

, O........MONO

O.........MNNO

M........MNNO

Q P R

Psin

Qsin

Qcos

Pcos

Qtan

Ptan

สรุปได้ว่า เมื่อ ABC เป็นสามเหลี่ยมมุมฉากท่ีมี a เป็นความยาวของด้านตรงข้ามมุม A , b เป็นความยาวของด้านตรงข้ามมุม B และ c เป็นความยาวของด้านตรงข้ามมุม C B ไซน ์(sine)ของมุม A คือ ……………………………………… c a โคไซน์ (cosine)ของมุม A คือ ……………..…….………………… แทนเจนต(์tangent) ของมุม A คือ ………………….….……………… A b C



ค าชี้แจง 1. ในรูปสามเหลี่ยมมุมฉากท่ีก าหนดให้แต่ละรูป จงหา sin A ,cos A ,tan A

รูปสามเหลี่ยมมุมฉาก วิธีท า หาค่าของ

sin A =…………………… cosA = …………………… tan A = …………………..

sin A =…………………… cosA = …………………… tan A = …………………..

sin A =…………………… cosA = …………………… tan A = …………………..

sin A =…………………… cosA = …………………… tan A = …………………..

sin A =…………………… cosA = …………………… tan A = …………………..

B 10 10 10 10

C 6 A C

C

A

B

11

61

A

B

C

15 8

C

B

A 28

21

A B

C

17

8

10



3. จงหาค่า ไซน์ โคไซน์ และแทนเจนต์ของมุม A และมุม B จากรูปต่อไปนี้

1) sin A =………… cos A =………. sin B =………… cos B =………. tan A =………… tan B =………. 2) sin A =………… cos A =………. sin B =………… cos B =………. tan A =………… tan B =………. 3) sin A =………… cos A =………. sin B =………… cos B =………. tan A =………… tan B =……….

3. จงหาว่าอัตราส่วนตรีโกณมิติที่ก าหนดให้ต่อไปนี้ เป็นอัตราส่วนตรีโกณมิติของมุม A หรือ B

1) sin………. = AB

BC

2) sin………. = AB

AC

3) cos……… = AB

AC

4) cos……… = AB

BC

5) tan……… = BC

AC

6) tan……… = AC

BC

AQ

BQ

CQ

12

13

5

4

CQ

AQ

BQ

AQ B

Q

CQ

3 4

B

A

C



4. จงหาว่าอัตราส่วนที่ก าหนดให้ต่อไปนี้ เป็นค่าไซน์หรือโคไซน์ของมุมที่ก าหนดให้

1) ………..A = c

a 2) …………A = c

b

3) ………..B = c

a 4) …………B = c

b

5. จากรูปสามเหลี่ยมมุมฉาก ABC ที่ก าหนดให้ จงหา

1) sin A = …………………….. 2) tan A = …………………….. 3) cos C = …………………….. 4) tan C = ……………………..

6. ก าหนดรูปสามเหลี่ยมมุมฉาก ABC ซึ่งมีมุม C เป็นมุมฉากและมีความยาวของด้านดังรูป จงหาอัตราส่วนตรีโกณมิติ ไซน์ โคไซน์ และ แทนเจนต์ ของ มุม A และ B

วิธีท า จากที่ก าหนดให้ AC = 12 หน่วย BC = 5 หน่วย

tanA = ACBC = ……………………….

tanB = …………………………………. ดังนั้น Asin ………………………………………………..

sinB = ………………………………………………… cosA = ………………………………………………..

cosB = ………………………………………………..

B

C

c

a b

A

A B

C

9

15

B

A C



Trigonometric Functions: The Unit Circle

The two historical perspectives of trigonometry incorporate different methods for introducing the trigonometric functions. Our first introduction to these functions is based on the unit circle.

Consider the unit circle given by วงกลมเป็นกราฟของความสัมพันธ์ {(x,y) x2 + y2 = 1} ก าหนดจ านวนจริง (ทีตา) วัดระยะจากจุด (1,0) ไป

ตามส่วนโค้งของวงกลมหนึ่งหน่วยยาว หน่วย ถึงจุด (x,y) ซึ่งอยู่บนวงกลมหนึ่งหน่วย โดยที่ 1. > 0 วัดส่วนโค้งจากจุด (1,0) ไปในทิศทางทวนเข็มนาฬิกา 2. < 0 วัดส่วนโค้งจากจุด (1,0) ไปในทิศทางตามเข็มนาฬิกา 3. = 0 จุดปลายส่วนโค้งคือจุด (1,0)

Imagine that the real number line is wrapped around this circle, with positive

numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure 4.21.

> 0 < 0



The Trigonometric Functions

From the preceding discussion, it follows that the coordinates x and y are two

functions of the real variable t. You can use these coordinates to define the six

trigonometric functions of t

sine cosecant cosine secant tangent cotangent These six functions are normally abbreviated sin, csc, cos, sec, tan, and cot, respectively.

รูปสามเหลี่ยมด้านเท่า ABC ให้ BC เป็นฐานยาว 2 หน่วย

sin cos tan csc sec cot 30

60

B

2

A

2

C D 1

60o 60o

30o



รูปสามเหลี่ยมมุมฉาก หน้าจั่ว ABC

sin cos tan csc sec cot 45

Radian and Degree Measure

B

C

A 45o

45o

1

1



Recall that the four quadrants in a coordinate system are numbered I, II, III, and

IV. Figure 4.8 on page 2 shows which angles between 0 and 2 lie in each of the four quadrants. Note that angles between 0 and /2 are acute angles and angles between

and are obtuse angles.



Degree Measure

A second way to measure angles is in terms of degrees, denoted by the

symbol ( ๐) A measure of one degree ( 1๐ ) is equivalent to a rotation of 360

1of a

complete revolution about the vertex. To measure angles, it is convenient to mark degrees on the circumference of a circle, as shown in Figure 4.13. So, a full revolution (counterclock-wise) corresponds to 360 ๐ , a half revolution to 180๐ a quarter revolution to 90๐ and so on. Because 2 radians corresponds to one complete revolution, degrees and radians are related by the equations 360 = 2 rad and 180 = rad From the latter equation, you obtain

1 =

rad and 1 rad =

π

180

which lead to the conversion rules .



Example : Converting from Degrees to Radians

1. 135 = ……………………………………………………………………………..

2. 540 = ……………………………………………………………………………..

3. - 270 = ……………………………………………………………………………..

And converting from Radians to Degrees

4. -

rad = ……………………………………………………………………………..

5. -

rad = ……………………………………………………………………………..

6. 2 rad = ……………………………………………………………………………..



In Figure 4.22, the unit circle has been divided into eight equal arcs, corresponding to t -values of

Similarly, in Figure 4.23, the unit circle has been divided into 12 equal arcs, corresponding to t -values of



Example : Evaluating Trigonometric Functions

Evaluate the six trigonometric functions at each real number.

sin cos tan csc sec cot

Degrees radians 0

30

45

60

90

120

135

180

210

225

240

270

300

315

330

360

y

x



Example : Evaluating Trigonometric Functions

Evaluate the six trigonometric functions at each real number.

sin cos tan csc sec cot

Degrees radians 0

- 30

- 45

-60

-90

-120

-135

-180

-210

-225

-240

-270

-300

-315

-330

-360

y

x



WORK . ให้นักเรียนเติมตารางให้สมบูรณ์

sin cos tan csc sec cot radians Degrees

2

4

3

65

43

34

35

32

47

611

23

2



WORK . ให้นักเรียนเติมตารางให้สมบูรณ์

sin cos tan csc sec cot Negative Positive

-2

-4

-

-3

-6

5

-4

3

-3

4

-3

5

-3

2

-4

7

-6

11

-2

3

-2



Example : ให้นักเรียนหาค่าของ sin และ cos เมื่อ เป็นจ านวนจริง เขียนจ านวนจริง ในรูป 2n เมื่อ n เป็นจ านวนเต็ม และ 0 2 ข้อที่ จ านวนจริง แนวคิด sin cos 1 5 2 -3

3 - 2

7

4 63

5 - 47

6 2

27

7 -2

13

8 31

9 -67

10 4

9

11 2

9

12 3

13

13 6

19

14 4

21

15 -6

23

16 -3

29

17 3

100



Domain and Period of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle shown in Figure 4.24. By definition, sin t = y and cos t = x. Because (x, y) is on the unit circle, you know that -1 y 1 and -1 x 1. So, the values of sine and cosine also range between -1 and 1. Adding 2 to each value of t in the interval [0, 2] completes a second evolution around the unit circle, as shown in Figure 4.25. The values of sin(t + 2) and cos(t + 2) correspond to those of sin t and cos t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result

sin (t + 2n) = sin t and cos (t + 2n) = cos t for any integer n and real number t. Functions that behave in such a repetitive (or cyclic) manner are called periodic.



Example : Using the Period to Evaluate the Sine and Cosine

1. sin 4 = ……………………………………………………………………………………………………..

2. cos 3 = ……………………………………………………………………………………………………..

3. cos 3

7= ……………………………………………………………………………………………………..

4. sin 4

9= ……………………………………………………………………………………………………..

5. cos4

17= ……………………………………………………………………………………………………..

6. sin6

19= ……………………………………………………………………………………………………..

7. sin (- 3

8)= …………………………………………………………………………………………………

8. cos (- 4

9)= …………………………………………………………………………………………………



Exercises : use the value of the trigonometric function using its period as an aid. trigonometric function to evaluate the indicated functions.

1. sin = 21

a. sin (- ) = …………………………………… b. csc (- ) = ……………………………………

2. sin (-) = 83

a. sin = …………………………………… b. csc = ……………………………………

3. cos (-) = -51

a. cos = …………………………………… b. sec (-) = ……………………………………

4. cos = - 43

a. cos (- ) = …………………………………… b. sec (- ) = ……………………………………

5. sin = 54

a. sin ( - ) = …………………………………… b. sin ( + ) = ……………………………………

6. cos = 54

a. cos ( - ) = …………………………………… b. cos ( + ) = …………………………………



Work : ให้นักเรียนหาค่าของฟังก์ชันตรีโกณมิติของจ านวนจริง 1. sin 0.495 …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… ……………………………………………………………………

2. sin 0.8862 …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… ……………………………………………………………………

3. cos 1 …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… ……………………………………………………………………

4. cos 0.33 …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… ……………………………………………………………………

5. tan 0.5575 …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… ……………………………………………………………………

6. tan 1.5 …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… ……………………………………………………………………



Basic Sine and Cosine Curves In this section, you will study techniques for sketching the graphs of the sine and cosine functions. The graph of the sine function is a sine curve. In Figure 4.47, the black portion of the graph represents one period of the function and is called one cycle of the sine curve. The gray portion of the graph indicates that the basic sine curve repeats indefinitely in the positive and negative directions. The graph of the cosine function is shown in Figure 4.48. Recall from Section 4.2 that the domain of the sine and cosine functions is the set of all real numbers. Moreover, the range of each function is the interval [-1,1] and each function has a period of 2. Do you see how this information is consistent with the basic graphs shown in Figures 4.47 and 4.48? Note in Figures 4.47 and 4.48 that the sine curve is symmetric with respect to the origin, whereas the cosine curve is symmetric with respect to the y-axis. These properties of symmetry follow from the fact that the sine function is odd and the cosine function is even.

GRAPHS OF SINE AND COSINE FUNCTIONS



To sketch the graphs of the basic sine and cosine functions by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points, and minimum points (see Figure 4.49). Example : จงเขียนกราฟต่อไปนี้

1. y = sin x

x 0

y

2. y = cos x

x 0

y



Example 1 Using Key Points to Sketch a Sine Curve Sketch the graph of y = 2 sin x on the interval [- , 4]

x y



x

y = cos x y=

cos x

y= 3 cos x



The amplitude is ……………………………………………………………………………………………. Period =……………………………………………………………………………………………………….…….

x y = sin x

y= sin

Work : Sketch the graph of y = sin

, y= cos

x y= sin

y= cos



……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 6 Vertical Translation

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Work : In Exercises , find the period and amplitude.

สมการ คาบ

(period) amplitude. (แอมพลิจูด) กราฟ

1.

2.

3.



สมการ คาบ

(period) amplitude. (แอมพลิจูด) กราฟ

4.

5.

6.

7.

8.

9.

10.

11. sketch the graph of the function. (Include two full periods.) ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Work : เขียนค่า a , b , c , d พร้อมเส้นกราฟท่ีต่างกันของฟังก์ชันตรีโกณมิติที่ก าหนดและตอบค าถาม ข้อ y=a sin (bx –c ) + d a b c d วาดกราฟท่ีได้ 1 y = sin x y = 2sin x y = 3sin x y =

sin x

2 y = sin (x) y = sin (

)

y = sin (2x) y = sin (3x) 3 y = sin (x) y = sin (x -

)

y = sin (x +

)

4 y = sin (x) y = sin (x) + 1 y = sin (x) + 2 y = sin (x) - 1

เติมค าตอบ 1. จากตารางข้อ 1 ค่าท่ีท าให้กราฟแตกต่างกัน คือค่า …………… (a,b,c,d) และท าให้กราฟต่างกันคือ

………………………………………………………………………………………………………………………………………. 2. จากตารางข้อ 2 ค่าท่ีท าให้กราฟแตกต่างกัน คือค่า …………… (a,b,c,d) และท าให้กราฟต่างกันคือ



……………………………………………………………………………………………………………………………………….



Work : เขียนค่า a , b , c , d พร้อมเส้นกราฟท่ีต่างกันของฟังก์ชันตรีโกณมิติที่ก าหนดและตอบค าถาม ข้อ y=a cos (bx –c ) + d a b c d วาดกราฟท่ีได้ 1 y = cos x y = 2 cos x y = 3 cos x y =

cos x

2 y = cos (x) y = cos (

)

y = cos (2x) y = cos (3x) 3 y = cos (x) y = cos (x -

)

y = cos (x +

)

4 y = cos (x) y = cos (x) + 1 y = cos (x) + 2 y = cos (x) - 1

เติมค าตอบ 1. จากตารางข้อ 1 ค่าท่ีท าให้กราฟแตกต่างกัน คือค่า …………… (a,b,c,d) และท าให้กราฟต่างกันคือ




……………………………………………………………………………………………………………………………………….



Work : In Exercises 1– 4, find a and d for the function f(x) = a cos x + d such that the graph of f matches the figure ข้อ กราฟ a d f(x) = a cos x + d 1

2

3

4



Work : In Exercises 1– 4, find a , b and c for the function f(x) = a sin (bx – c) such that the graph of f matches the figure ข้อ กราฟ a b c f(x) = a sin (bx – c) 1

2

3

4



GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS



Example : จงเขียนกราฟต่อไปนี้ y = tan x

x

0

y

Example 1 : Sketching the Graph of a Tangent Function Sketch the graph of

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 2 : Sketching the Graph of a Tangent Function Sketch the graph of ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example : จงเขียนกราฟต่อไปนี้ y = cot x

x

0

y



Example 3 : Sketching the Graph of a Cotangent Function Sketch the graph of

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example : จงเขียนกราฟต่อไปนี้ y = csc x

x

0

y

Example : จงเขียนกราฟต่อไปนี้ y = sec x

x

0

y



Example 4 : Sketching the Graph of a Cosecant Function Sketch the graph of

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 5 : Sketching the Graph of a Secant Function Sketch the graph of ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Work : ก าหนดค่า a , b , c , d พร้อมวาดกราฟของฟังก์ชันท่ีก าหนดและเติมค าตอบให้สมบูรณ์ ข้อ ฟังก์ชัน วาดกราฟที่ได้ ฟังก์ชัน sin ฟังก์ชัน csc 1

โดเมน คือ โดเมน คือ

เรนจ์ คือ


แอมพลิจูด =


คาบ =

คาบ =

2






คาบ =

คาบ =

3






คาบ =

คาบ =



ข้อ ฟังก์ชัน วาดกราฟที่ได้ ฟังก์ชัน sin ฟังก์ชัน csc 4






คาบ =

คาบ =

5






คาบ =

คาบ =

เขียนการเปลี่ยนแปลงของแต่ละฟังก์ชัน เมื่อค่า a , b, c , d เปลี่ยนแปลง 1. ค่า a ท าให้กราฟ sine ………………………………………….. กราฟ csc……………………………………….

2. ค่า b ท าให้กราฟ sine ………………………………………….. กราฟ csc……………………………………….

3. ค่า c ท าให้กราฟ sine ………………………………………….. กราฟ csc……………………………………….

4. ค่า d ท าให้กราฟ sine ………………………………………….. กราฟ csc……………………………………….



Work : ก าหนดค่า a , b , c , d พร้อมวาดกราฟของฟังก์ชันท่ีก าหนดและเติมค าตอบให้สมบูรณ์ ข้อ ฟังก์ชัน วาดกราฟที่ได้ ฟังก์ชัน cos ฟังก์ชัน sec 1






คาบ =

คาบ =

2






คาบ =

คาบ =

3






คาบ =

คาบ =



ข้อ ฟังก์ชัน วาดกราฟที่ได้ ฟังก์ชัน cos ฟังก์ชัน sec 4






คาบ = คาบ =

5






คาบ =

คาบ =

เขียนการเปลี่ยนแปลงของแต่ละฟังก์ชัน เมื่อค่า a , b, c , d เปลี่ยนแปลง 1. ค่า a ท าให้กราฟ cosine ……………………………………….. กราฟ secant ……………………………….

2. ค่า b ท าให้กราฟ cosine ………………………………………. กราฟ secant ……………………………….

3. ค่า c ท าให้กราฟ cosine ……………………………………….. กราฟ secant ……………………………….

4. ค่า d ท าให้กราฟ cosine …………………………………….. กราฟ secant ……………………………….



Work : ก าหนดค่า a , b , c , d พร้อมวาดกราฟของฟังก์ชันท่ีก าหนดและเติมค าตอบให้สมบูรณ์ ข้อ ฟังก์ชัน วาดกราฟที่ได้ ฟังก์ชัน tan ฟังก์ชัน cot 1






คาบ =

คาบ =

2






คาบ =

คาบ =

3






คาบ =

คาบ =



ข้อ ฟังก์ชัน วาดกราฟที่ได้ ฟังก์ชัน tan ฟังก์ชัน cot 4






คาบ = คาบ =

5






คาบ =

คาบ =

เขียนการเปลี่ยนแปลงของแต่ละฟังก์ชัน เมื่อค่า a , b, c , d เปลี่ยนแปลง 1. ค่า a ท าให้กราฟ tangent …………………………………….. กราฟ cotangent …………………………

2. ค่า b ท าให้กราฟ tangent …………………………………….. กราฟ cotangent …………………………

3. ค่า c ท าให้กราฟ tangent …………………………………….. กราฟ cotangent …………………………

4. ค่า d ท าให้กราฟ tangent …………………………………….. กราฟ cotangent …………………………



SUM AND DIFFERENCE FORMULAS

Using Sum and Difference Formulas

In this and the following section, you will study the uses of several trigonometric identities and formulas.

Example1 Evaluating a Trigonometric Function

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 2 Evaluating a Trigonometric Function

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 3 Evaluating a Trigonometric Function

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 4 Proving a Cofunction Identity

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 5 Deriving Reduction Formulas

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



VOCABULARY: Fill in the blank.

……………………………………………………………………………………………………………

……………………………………………………………………………………………………………

……………………………………………………………………………………………………………

……………………………………………………………………………………………………………

……………………………………………………………………………………………………………

…………………………………………………………………………………………………………… SKILLS AND APPLICATIONS In Exercises 7–12, find the exact value of each expression.

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



In Exercises 13 – 20, write the expression as the sine, cosine,or tangent of an angle. 13. sin 3 cos 1.2 - cos 3 sin 1.2 =………………………………………………………

14. =……………………………………………..………

15. =………………………………………………………

16. =………………………………………………………

17. =………………………………………………………

18. =………………………………………………………

19. =………………………………………………………

20. =……………………………………………………… In Exercises 21– 27, find the exact value of the expression.

21.

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

22.

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… 23 ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



25 ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

26

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

27

……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… In Exercises 28–35, find the exact value of the trigonometric function given that

and (Both and are in Quadrant II.)

28 29

…………………………………………………………… …………………………………………………………… …………………………………………………………… ……………………………………………………………

30 31 …………………………………………………………… …………………………………………………………… …………………………………………………………… ……………………………………………………………

32 33 …………………………………………………………… …………………………………………………………… …………………………………………………………… …………………………………………………………… …………………………………………………………… ……………………………………………………………

34 35 …………………………………………………………… …………………………………………………………… …………………………………………………………… ……………………………………………………………



Multiple-Angle Formulas In this section, you will study four other categories of trigonometric identities. 1. The first category involves functions of multiple angles such as sin ku and cos ku 2. The second category involves squares of trigonometric functions such as sin2 u 3. The third category involves functions of half-angles such as sin (u/2) 4. The fourth category involves products of trigonometric functions such as sin u cos v

Example 1 Solving a Multiple-Angle Equation

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

MULTIPLE-ANGLE AND PRODUCT-TO-SUM FORMULAS



Example 2 Using Double-Angle Formulas to Analyze Graphs

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

x

y

Example 3 Evaluating Functions Involving Double Angles

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 4 Deriving a Triple-Angle Formula Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

In Exercises 1–8, use the figure to find the exact value of the trigonometric function.

1 2 …………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

3 4 …………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

5 6 …………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

7 8 …………………………………………………………… …………………………………………………………… …………………………………………………………… ……………………………………………………………



Power-Reducing Formulas The double-angle formulas can be used to obtain the following power-reducing formulas. Example 5 shows a typical power reduction that is used in calculus.

Example 5 Reducing a Power

Solution Note the repeated use of power-reducing formulas. ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

Half-Angle Formulas You can derive some useful alternative forms of the power-reducing formulas by replacing with u/2. The results are called half-angle formulas.



Example 6 Using a Half-Angle Formula

Find the exact value of sin 105

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 7 Solving a Trigonometric Equation

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 8 Writing Products as Sums

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 9 Using a Sum-to-Product Formula

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 10 Solving a Trigonometric Equation

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 11 Verifying a Trigonometric Identity

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

In Exercises 11 –15, find the exact solutions of the equation in the interval

[0, 2). 11.

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



12. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… 13.

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… 14.

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

15. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



In Exercises 16– 18, find the exact values of sin 2 , cos 2 and tan 2 using the double-angle formulas.

16.

Solution

sin 2 cos 2 tan 2 ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… …………………………………………

17.

Solution

sin 2 cos 2 tan 2 ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… …………………………………………

18.

Solution

sin 2 cos 2 tan 2 ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… …………………………………………



In Exercises 19–22, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. 19. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

20. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

21. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

22. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



In Exercises 23–28, use the figure to find the exact value of the trigonometric function.

23

24

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

25

26

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

27

28

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………

…………………………………………………………… ……………………………………………………………



In Exercises 29–31, (a) determine the quadrant in which /2 lies, and (b) find

the exact values of sin (/2 ) , cos (/2 ) and tan (/2 ) using the half-angle formulas

29.

Solution ………………………………………………………………………………………………………………………………………………

sin (/2 ) cos (/2 ) tan (/2 ) ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… …………………………………………

30.

Solution ………………………………………………………………………………………………………………………………………………

sin (/2 ) cos (/2 ) tan (/2 ) ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… …………………………………………

31.

Solution ………………………………………………………………………………………………………………………………………………

sin (/2 ) cos (/2 ) tan (/2 ) ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… ………………………………………… ……………………………………………… …………………………………………… …………………………………………



In Exercises 32 –35 use the sum-to-product formulas to find the exact value of the expression.

32. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

33. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

34. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

35. Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



INVERSE TRIGONOMETRIC FUNCTIONS



Example 1 Evaluating the Inverse Sine Function

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 2 Graphing the Arcsine Function

y

x = sin y



Example 3 Evaluating Inverse Trigonometric Functions

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 4 Using Inverse Properties

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 5 Evaluating Compositions of Functions

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 6 Some Problems from Calculus

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SKILLS AND APPLICATIONS In Exercises 4 –19, evaluate the expression without using a calculator.

4 …………………………………………………………………………………………………

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8

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In Exercises 20 and 21, use a graphing utility to graph f , g and y = x in the same viewing window to verify geometrically that g is the inverse function of (Be sure to restrict the domain of properly.)

20

x

f(x) g(x)



21

x f(x) g(x)

In Exercises 22 –25, use the properties of inverse trigonometric functions to evaluate the expression.

22 23 …………………………………………………………… ……………………………………………………………

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VERIFYING TRIGONOMETRIC IDENTITIES




Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 2 Verifying a Trigonometric Identity

Solution ………………………………………………………………………………………………………………………………………………

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Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 4 Converting to Sines and Cosines

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Example 6 Working with Each Side Separately

Solution ………………………………………………………………………………………………………………………………………………

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In Exercises 1 – 6, fill in the blank to complete the trigonometric identity.

1.

2.

3. 4.

5.

6.

In Exercises 7 –10, verify the identity. 7.

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

8.

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… 9.

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



10.

Solution ………………………………………………………………………………………………………………………………………………

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SOLVING TRIGONOMETRIC EQUATIONS



Example 1 Collecting Like Terms

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 2 Extracting Square Roots

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 3 Factoring

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 4 Factoring an Equation of Quadratic Type

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 5 Rewriting with a Single Trigonometric Function

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 6 Squaring and Converting to Quadratic Type

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 7 Functions of Multiple Angles

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 8 Functions of Multiple Angles

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 9 Using Inverse Functions

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



In Exercises 1–24, solve the equation 1.

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… 2.

Solution ………………………………………………………………………………………………………………………………………………

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6. Solution ………………………………………………………………………………………………………………………………………………

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7.

Solution ………………………………………………………………………………………………………………………………………………

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Solution ………………………………………………………………………………………………………………………………………………

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9. Solution ………………………………………………………………………………………………………………………………………………

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LAW OF SINES



Example 1 Given Two Angles and One Side—AAS

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 2 Given Two Angles and One Side—ASA

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



The Ambiguous Case (SSA) In Examples 1 and 2, you saw that two angles and one side determine a unique triangle. However, if two sides and one opposite angle are given, three possible situations can occur: (1) no such triangle exists, (2) one such triangle exists, or (3) two distinct triangles may satisfy the conditions.

Example 3 Single-Solution Case—SSA

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 4 No-Solution Case—SSA

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 5 Two-Solution Case—SSA

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 6 Finding the Area of a Triangular Lot

Solution ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 7 An Application of the Law of Sines

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

LAW OF COSINES



Example 8 Three Sides of a Triangle—SSS

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

Example 9 Two Sides and the Included Angle—SAS

Solution ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Applications Example 10 An Application of the Law of Cosines

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… Example 11 An Application of the Law of Cosines

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



Example 12 สมชายขับเรือไปทางทิศเหนือด้วยความเร็ว 16 กิโลเมตรต่อชั่วโมง เมื่อเวลา 11.00 น. เขาสังเกตเห็นเรืออีกล าหนึ่งจอดอยู่ทางทิศตะวันออก ซึ่งอยู่ในแนวเดียวกับประภาคาร และเมื่อ

เวลาผ่านไป 2 ชั่วโมง เขาพบว่าเรือล าที่จอดอยู่ในทิศ 145 และประภาคารอยู่ในทิศ 120 จงหาว่าเรือนั้นจอดอยู่ห่างประภาคารเท่าไร Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

Example 13 Using Heron’s Area Formula

Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………



In Exercises 1–5, use the Law of Sines to solve the triangle.Round your answers to two decimal places.

1.

………………………………………………………………. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………

2.

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3.

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4.

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5.

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In Exercises 6–34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. 6. ……………………………………………………………….

………………………………………………………………. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………

7. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………

8. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………

In Exercises 9– 10, find the area of the triangle having the indicated angle and sides. 9. ……………………………………………………………….

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10. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………. ………………………………………………………………. ……………………………………………………………… ………………………………………………………………



In Exercises 11–15, use the Law of Cosines to solve the triangle. Round your answers to two decimal places.

11.

…………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..………….

12.

…………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..………….

13.

…………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..………….

14.

…………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..………….

15. …………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..………….

In Exercises 16–17, determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle. Then solve the triangle. 16. …………………………………………………………………………..………….

…………………………………………………………………………..…………. …………………………………………………………………………..………….

17

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In Exercises 18– 19, use Heron’s Area Formula to find the area of the triangle. 18. …………………………………………………………………………..………….

…………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..………….

19.

…………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..…………. …………………………………………………………………………..………….

20. ชายคนหนึ่งยืนห่างจากตึกหลังหนึ่ง 20 เมตร มองเห็นยอดตึกและเสาอากาศซึ่งอยู่บนยอดตึกเป็นมุมเงย 30 และ 60 ตามล าดับ จงหาความสูงของเสาอากาศ (23.09 เมตร) Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

21. สายชลยืนอยู่บนหน้าผาสูง 100 ฟุต เห็นกองหินสองกองบนพ้ืนดินเบื้องล่างในแนวเดียวกัน ถ้ามุมก้มของกองหินท้ังสองเป็น 60 และ 45 ตามล าดับ จงหาว่าหินสองกองนั้นอยู่ห่างกันเท่าไร (73.2 ฟุต) Solution ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………

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