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Right Triangle Trigonometry

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Angles

Trigonometry: measurement of triangles

Angle Measure

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In this section, we will be studying special ratios of the sides of a right triangle, with respect to angle, .

These ratios are better known as our six basic trig functions:

Sine of

Cosine of

Tangent of

Cosecant of

Secant of Cotangent of

Trigonometric Functions

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Take a look at the right triangle, with an acute angle, , in the figure below.

Notice how the three sides are labeled in reference to .

The sides of a right triangle

Side adjacent to

S

ide

op

po

site

Hypotenuse

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Definitions of the Six Trigonometric Functions

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To remember the definitions of Sine, Cosine and Tangent, we use the acronym :

“SOH CAH TOA”

Definitions of the Six Trigonometric Functions

O A O

H HS C

AT

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Find the exact value of the six trig functions of :

Example

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9 First find the length of the hypotenuse using the Pythagorean Theorem.

2

2

2

hyp

hyp

hyp

hyp

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Example (cont)

5

9

106

So the six trig functions are:

sin

cos

tan

opp

hyp

adj

hyp

opp

adj

csc

sec

cot

hyp

opp

hyp

adj

adj

opp

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Given that is an acute angle and , find the exact value of the six trig functions of .

Example

12cos

13

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Find the value of sin given cot = 0.387, where is

an acute angle. Give answer to three significant digits.

Example

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The 45º- 45º- 90º Triangle

Special Right Triangles

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12

45º

45º

Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 45

sin 45 = csc 45 =

cos 45 = sec 45 =

tan 45 = cot 45 =

Ratio of the sides:

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The 30º- 60º- 90º Triangle

Special Right Triangles

Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 30

sin 30 = csc 30 =

cos 30 = sec 30 =

tan 30 = cot 30 =

1

3

60º

30º

2

Ratio of the sides:

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The 30º- 60º- 90º Triangle

Special Right Triangles

Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 60

sin 60 = csc 60 =

cos 60 = sec 60 =

tan 60 = cot 60 =

1

3

60º

30º

2

Ratio of the sides:

14

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MAKE SURE THE MODE IS SET TO THE CORRECT UNIT OF ANGLE MEASURE (i.e. Degree vs. Radian)

Example:

Find to three significant digits.

Using the calculator to evaluate trig functions

tan 46.2

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For reciprocal functions, you may use the button, but DO NOT USE THE INVERSE FUNCTIONS (e.g. )!

Example:

1. Find 2. Find

(to 3 significant dig) (to 4 significant dig)

Using the calculator to evaluate trig functions

csc73.2 cot 11.56

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THE INVERSE TRIG FUNCTIONS GIVE THE MEASURE OF THE ANGLE IF WE KNOW THE VALUE OF THE FUNCTION.

Notation:The inverse sine function is denoted as sin-1x.

It means “the angle whose sine is x”.

The inverse cosine function is denoted as cos-1x. It means “the angle whose cosine is x”.

The inverse tangent function is denoted as tan-1x. It means “the angle whose tangent is x”.

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Examples

Evaluate the following inverse trig functions using the

calculator. Give answer in degrees.

1 11. tan 1.372 2. sin 0.64

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Examples

Evaluate the following inverse trig functions using the

calculator. Give answer in degrees.

1 13. tan 1 4. cos 0.541

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Angles and Accuracy of Trigonometric Functions

Measurement of Angle to Nearest

Accuracy of Trig Function

1° 2 significant digits

0. 1° or 10’ 3 significant digits

0. 01° or 1’4 significant digits

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Example

Solve for y:

Solution:

y

52º

9.6

Since you are looking for the side adjacent to 52º and are given the hypotenuse, you should use the _____________ function.

WARNING: Make sure your MODE is set to “Degree”

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Example

Solve the right triangle with the indicated measures.

1. 40.7 8.20A a in

Solution

A= 40.7°

C B

b c

a=8.2”

Answers:

B

b

c

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Example

2. 25 35a c A

C B

b c=35

a=25

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Example

3. Find the altitude of the isosceles triangle below.

36°

8.6 m

36°

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Example

4. Solve the right triangle with 8.60 11.25a cm b cm

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Angle of Elevation and Angle of Depression

The angle of elevation for a point above a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point.

The angle of depression for a point below a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point.

Horizontal line

Horizontal line

Angle of elevation

Angle of depression

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Example

A guy wire of length 108 meters runs from the top of an

antenna to the ground. If the angle of elevation of the top

of the antenna, sighting along the guy wire, is 42.3° then

what is the height of the antenna? Give answer to three

significant digits.

Solution

108 m

42.3°

y

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