Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half...
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Transcript of Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half...
Ch4.1A – Radian and Degree Measure
r
Ch4.1A – Radian and Degree Measure
s ~3.14 arcs θ
= half circle r
θ = 1 radian
One radian – the measure of the angle when the arc length = the radius
1 revolution (360˚) = 2π radians (~6.28 arc lengths)
½ revolution ( ) = radians
¼ revolution ( ) = radians
1/3 revolution ( ) = radians
1/8 revolution ( ) = radians
Ch4.1A – Radian and Degree Measure
s θ r
θ = 1 radian
One radian – the measure of the angle when the arc length = the radius
1 revolution (360˚) = 2π radians (~6.28 arc lengths)
½ revolution (180˚) = π radians
¼ revolution (90˚) = radians
1/3 revolution (60˚) = radians
1/8 revolution (45˚) = radians
2
3
4
Ex1) Find the acute angle equivalent to
Ex2) Find the negative angle equivalent to
Ex3) Find the positive angle equivalent to
6
13
4
3
3
2
Ex4) Find the complement and supplement angles to a)
b)
5
2
5
4
Degree/Radian Conversions
Degree/Radian Conversions
Conversions: Conversion Factor:
rad 00
rad 2360
rad 2
90
rad 6
30
rad 3
60
rad 4
45
rad 180
rad 2
3270
rad 180
rad 180
1
rad
180or
180
rad
Ex5) Convert:a) 135˚
b) -270˚
c)
d) 2 rad
(Quiz on conv in 2 days) Ch4.1A p318 5-19odd,45-55odd
rad 2
Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)
Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)
Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)
Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)
Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)
Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)
Ch4.1B – Arc Length(Quiz tomorrow!)
r
Ch4.1B – Arc Length
r
Length of a circular arc: s = r.θ (θ must be in radians)
Ex1) A circle has a radius of 4inches. What is the arc length intercepted by a central angle of 240˚.
Linear speed: distance traveled Angular speed: angle swept out
time time
omega (θ must be in radians)
Ex3) The second hand of a clock is 10.2cm long.Find the speed of the second hand.
t
t
dv
Ex3) A lawn roller is 30in in diameter and makes 1 revolutionevery 5/6 sec.a) Find the angular speedb) How fast does it move across the lawn?
Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Do #35 and #39 in class)(Quiz tomorrow on conversions!)
Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)
Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)
Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)
Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)
Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)
Ch4.1 Quiz Name___________Convert radians to degrees: A B C D
Convert degrees to radians: A B C D 4. 270˚ 4. 90˚ 4. 180˚ 4. 360˚ 5. 60˚ 5. 30˚ 5. 120˚ 5. 150˚ 6. 210˚ 6. 240˚ 6. 330˚ 6. 300˚
6
7 3.
3
4 3.
3
5 3.
6
11 3.
3
2. 6
2. 6
5 2.
3
2 2.
rad 2
3 1. rad
2 1. rad 2 1. rad .1
Ch4.2 – The Unit Circle
x2 + y2 = 1
Ex1) 45˚ = _____ rad
x = _____
y = _____
Ex2) 60˚ = _____ rad
x = _____
y = _____
Ex3) 30˚ = _____ rad
x = _____
y = _____ Ex5) 90˚ = _____ rad
Ex4) 0˚ = _____ rad x = _____
x = _____ y = _____
y = _____
Ex6)x = _____ x = _____y = _____ y = _____
x = _____ x = _____y = _____ y = _____
Trig Functions(sine) (cosine) (tangent)
sin t = y cos t = x tan t =
Ex7) Eval 3 trigs for:
a)
b)
c)
d)
Ch4.2A p328 1-39odd (only sin,cos,tan)
x
y
6
t
4
5t
t
2
3t
Ch4.2A p328 1-39odd (only sin,cos,tan)
Ch4.2A p328 1-39odd (only sin,cos,tan)
Ch4.2A p328 1-39odd (only sin,cos,tan)
Ch4.2A p328 1-39odd (only sin,cos,tan)
Ch4.2B – Trig Functions
sin t = y (cosecant) csc t =
cos t = x (secant) sec t =
tan t = (cotangent) cot t =
Ex8) Eval 6 trigs for:x
y
3
t
y
1
x
1
y
x
sin t = y csc t =
cos t = x sec t =
tan t = cot t =
Ex9) Eval:
a)
b)
x
y
6
13sin
y
1
x
1
y
x
2
7cos
x = cos t y = sin t
Domains: (what you put in for t)
Ranges: (what you get out for x or y)
Types of functions:
1. x = cos t is an even function
(So is secant)
2. y = sin t is an odd function
(So is tan, csc, and cot)
Ex10) Use calc:a) sin 76.4˚ (must be in degree mode.)b) cot 1.5 (must be in radian mode.)
Ch4.2B p328 57, 2-38 (eoe)
Ch4.2B p328 57, 2-38 (every other even)
Ch4.2B p328 57, 2-38 (every other even)
Ch4.2B p328 57, 2-38 (every other even)
Ch4.2B p328 57, 2-38 (every other even)
Ch4.2B p328 57, 2-38 (every other even)
Ch4.3 – Right Triangle TrigQuiz tomorrow on this!
SOH-CAH-TOA
sin θ = cos θ = tan θ =
Θhypotenuse
adjacent
opposite
Ch4.3 – Right Triangle Trig
SOH-CAH-TOA
sin θ = cos θ = tan θ =
csc θ = sec θ = cot θ =
Ex1) Eval 6 trigs for:
5 4
3
hyp
opp
Θhypotenuse
adjacent
opposite
hyp
adjadj
opp
opp
hyp
opp
hypopp
adj
Θ
Ch4.3 – Right Triangle Trig
SOH-CAH-TOA
sin θ = cos θ = tan θ =
csc θ = sec θ = cot θ =
Ex2) Find the value of sin45˚, cos45˚, tan45˚
hyp
opp
Θhypotenuse
adjacent
opposite
hyp
adjadj
opp
opp
hyp
opp
hypopp
adj
45˚
Ex3) Use the equilateral triangle to find the value of sin60˚, cos60˚, sin30˚, cos30˚
Sine, Cosine, and Tangent of Special Angles
sin30˚ = cos30˚ = tan30˚ =
sin45˚ = cos45˚ = tan45˚ = 1
sin60˚ = cos60˚ = tan60˚ =
HW#8) Find exact values of 6 trigs for:
3
6
Ch4.3A p338 1-22(a,b) Quiz tomorrow – would u like 2 c a sample?
2
1
2
1
2
3
2
3
3
3
3
2
2
2
2
Θ
Sine, Cosine, and Tangent of Special Angles
sin30˚ = cos30˚ = tan30˚ =
sin45˚ = cos45˚ = tan45˚ = 1
sin60˚ = cos60˚ = tan60˚ =
HW#8) Find exact values of 6 trigs for:
3
6
Ch4.3A p338 1-22(a,b) Quiz tomorrow – would u like 2 c a sample?
2
1
2
1
2
3
2
3
3
3
3
2
2
2
2
Θ
Trigonometry & Vector Components
SOHCAHTOA
in
pp
yp
os
dj
yp
an
pp
dj
sinΘ =
cosΘ =
tanΘ =
opphyp
adjhyp
oppadj
Ch4.3A p338 1-22(a,b) Quiz today!
Ch4.3A p338 1-22(a,b)
Ch4.3A p338 1-22(a,b)
Ch4.3A p338 1-22(a,b)
Ch4.2 Quiz Name___________Find exact values:˚ A B C D
1. sin 30˚ 1. cos 30˚ 1. sin 60˚ 1. cos 60˚2. tan 30˚ 2. tan 60˚ 2. sin 30˚ 2. cos 30˚ 3. cos 60˚ 3. sin 60˚ 3. tan 45˚ 3. tan 30˚4. tan 45˚ 4. cos 45˚ 4. cos 60˚ 4. sin 60˚5. sin 60˚ 5. cos 60˚ 5. tan 30˚ 5. tan 45˚ 6. cos 45˚ 6. tan 45˚ 6. cos 45˚ 6. sin 45˚
Ch4.3B – Trig Identities
Reciprocals:
sin θ = cos θ = tan θ =
csc θ = sec θ = cot θ =
Combos:
Pythag: Quiz in 2 days on these identities.
csc
1
sec
1cot
1
sin
1
cos
1tan
1
sin
coscot
cos
sintan
22
22
22
csccot1
sectan1
1cossin
Ex4) Let θ be acute angle, with sin θ = 0.6, find:a) cos θ b) tan θ
Ex5) Let θ be acute, with tan θ = 3, find:a) cot θ b) sec θ
θ˚
θ˚
Ex6) Use a calc to eval:a) cos 28˚b) sec 28˚c) sec 5˚40’
Ex7) Find the value of θ in radians and degrees:
a) sin θ = b) cos θ =
b) csc θ = 2
Ex8) Use calc to find θ in:a) degrees for cos θ = 0.3746b) radians for sin θ = 0.3746
Ch4.3B p339 23-31odd,37-40all(a only),47-55all(a only) Quiz in 2 days
2
32
2
Ch4.3B p339 23-31odd,37-40all(a only), 47-55all(a only)
Ch4.3B p339 23-31odd,37-40all(a only), 47-55all(a only)
Ch4.3B p339 23-31odd,37-40all(a only), 47-55all(a only)
Ch4.3C – Trig Word Problems (Quiz on ID’s tomorrow!) Ex7) A surveyor is standing 50ft from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5˚. How tall is the tree?
h = ?
θ=71.5˚| --------50ft-----------|
Ex8) A person is 200yds straight away from a river. The person walks at an angle, going 400yds til he gets to the river’s edge. At what angle did he walk?
200yds 400yds
Ex9) A 12 meter flagpole casts a shadow 9 meters long. What is the angle of elevation to the sun?
Ch4.3C p33924-32even,68-70all,81,82Quiz tomorrow. Would u like to c an example?
Quiz example:
=
=
csc
1
tan
1
cos
sin
2
2
sec_____1
1cos_____
Ch4.3C p33924-32even,68-70all,81,82
Ch4.3C p33924-32even,68-70all,81,82
Ch4.3C p33924-32even,68-70all,81,82
Ch4.3C p33924-32even,68-70all,81,82
Ch4.3 – Identities Quiz Name__________Reciprocals A B C D
1.
2.
Combinations
3.
Pythag
4.
5.
csc
1
tan
1
cos
sin
2sec___1
sec
1
cot
1
tan
1
cot
1
sin
1
cos
1
cos
sin
sin
cos
sin
cos
1cos___ 2 1___sin 2 ___cot1 2
1___sin 2 1cos___ 2 ___tan1 2 2csc___1
Ch4.4A – Trig Functions of Any Angle
(x,y)
Ch4.4A – Trig Functions of Any Angle
(x,y) sin θ = cos θ = tan θ = r θ
csc θ = sec θ = cot θ =
Reminder: Ex1) Let -3,4 be a point on the QII QI terminal side of an angle θ. sinθ = sinθ = find sin,cos,tan.cosθ = cosθ =tanθ = tanθ = QIII QIVsinθ = sinθ = cosθ = cosθ =tanθ = tanθ =
r
yr
x
x
y
y
r
x
r
y
x
Ex2) Given tan θ = . and cos θ > 0. Find sin θ and sec θ.
Ex3) Evaluate sin and tan at
4
5
2
3,,
2,0
Reference Angles- any given angle has an equivalent angle where0 > θ > 90˚ , or 0 > θ > π/2.
Ex4) Find ref angle:a) 300˚b) 2.3c) -135˚
Ex5) Evaluate:a) cos b) tan (-210˚)
c) csc
Quiz in 2 days (Ex?) Ch4.4A p349 2-42eoe (all 6 trigs)
3
4
4
11
Sample Quiz for Ch4.2/4.3
1.
2.
3.
4.
5.
6.
7.
8. )cos(
)300sin(
)270cos(
3
2sin
6
5cos
)135sin(
)135cos(
6
7sin
Lab4.1 – Heights and Lengths (angles measured)
Ch4.4A p349 2-42eoe (all 6 trigs)
Ch4.4A p349 2-42eoe (all 6 trigs)
Ch4.4A p349 2-42eoe (all 6 trigs)
Ch4.4A p349 2-42eoe (all 6 trigs)
Lab4.1 – Heights and Lengths (angles measured)
Ch4.4B – More Trigs at Any Angle
Ex6) Let θ be an angle in QII, such that sinθ = Find cos θ and tan θ using trig identities.
Ex7) Use a calculator to evaluate:a) cot (410˚)
b) sin (-7)
c) Solve for θ: tan θ = 4.812, where 0 < θ < 2π
Ch4.4B p350 43-81odd (a only) Quiz tomorrow!
3
1
Ch4.4B – More Trigs at any Angle
HW#43) Eval sin,cos,tan for:a) 225˚
#63) Find 2 values in degrees and radiansa) sin θ = ½ , where 0 < θ < 2π
Ch4.4B p350 43-81odd (a only) Quiz tomorrow!
Ch4.4B p350 43-81odd (a only)
Ch4.4B p350 43-81odd (a only)
Ch4.4B p350 43-81odd (a only)
Ch4.4B p350 43-81odd (a only)
Ch4.5A – Graphs of Sine and Cosine (Quiz first)Ex1) What are the values of sine and cosine at: θ sin θ cos θ
22
3
2
3
4
6
0
Ch4.5A – Graphs of Sine and Cosine (Quiz first)Ex1) What are the values of sine and cosine at: θ sin θ cos θ
Ex2) Graph on a # line: y = sin x
Ex3) Graph on a # line: y = cos x
1 0 2
0 1- 2
3
1- 0
0 1 2
2
1
2
3
3
2
2
2
2
4
2
3
2
1
6
1 0 0
y = a.sin x y = a.cos x
Amplitude – stretches and shrinks graph vertically
Ex4) Sketch y = 2.sin x
θ 2.sinx
22
3
2
0
y = a.sin x y = a.cos x
Amplitude – stretches and shrinks graph vertically
Ex5) Sketch y = ½.cos x
θ ½.cos x
22
3
2
0
Period of Sine and Cosine(Normally its 2π)
y = a.sin (bx) y = a.cos (bx)
b determines the period
Ex6) Sketch y = sin(½x) vs y = sin (x) vs y = sin(2x)
1
–1
Ch4.5A p361 1–13odd, 43,45
bperiod
2
4 3 2 2
3
2
Ch4.5A p361 1–13odd, 43,45
Ch4.5A p361 1–13odd, 43,45
Ch4.5A p361 1–13odd, 43,45
Ch4.5A p361 1–13odd, 43,45
Ch4.5B – Translations
y = a.sin(bx – c) y = a.cos(bx – c)
a = amplitude c = horizontally shifts b determines the period the period
start: bx – c = 0end: bx – c = 2π
Ex7) Sketch y = ½.sin(x – )
bperiod
2
3
2 2
3
2
Ex8) y = 3.cos(2πx + 1)Use a calc to find its period
Ex9) Sketch y = 2.cos(2x – π) + 1
Ch4.5B p361 15–21odd,47–53odd (lets do 17 and 47 in class)
Ch4.5B p361 15–21odd,47–53odd
Ch4.6A – Graphs of Other FunctionsEx1) Sketch the graph of y = tan x θ tan θ
4
2
2
3
2
4
0
Ch4.6A – Graphs of Other FunctionsEx1) Sketch the graph of y = tan x θ tan θ
To find asymptotes for tangent:0 2
undefined 2
3
0
undefined 2
1 4
0 0
2
2
cbx
cbx2
2
Ex2) Sketch the graph of y = tan To find asymptotes θ tan θ for tangent:
4
2
2
3
2
4
0
2
x
2
2
cbx
cbx
Ex3) Sketch the graph of y = –3tan2x To find asymptotes θ tan θ for tangent:
4
2
2
3
2
4
0
2
2
cbx
cbx
Ex4) Sketch the graph of y = cot x θ cot θ
4
2
2
3
2
4
0
sin
coscot
Ex5) Sketch the graph of y = 2cot To find asymptotes θ cot θ for cotangent:
Ch4.6A p372 9-12,21,22,24
4
2
2
3
2
4
0
cbx
cbx 0
3
x
Ch4.6A p372 9-12,21,22,24
Ch4.6B – Graphs of Other FunctionsEx6) Sketch the graph of y = csc x θ sin x csc x
To find asymptotes for cosecant:
anywhere sin x = 0
2
2
3
2
4
0
Ex7) Sketch the graph of y = sec x θ cos x sec x
To find asymptotes for secant:
anywhere cos x = 0
2
2
3
2
4
0
HW#13) Sketch the graph of y = -½sec x
HW#19) Sketch the graph of y = csc
Ch4.6B p372 13-20all,23,26
2
x
Ch4.6B p372 13-20all,23,26
Ch4.7 – Inverse Trig Functions
Ex1) Graph y = sin x
1
-1 2
2
3
2
2- -
Ch4.7 – Inverse Trig Functions
Ex1) Graph y = sin x
1
-1
The inverse function of sin x is called sin-1 x or arcsin x - its domain is [-1,1], and its range is .
Graph arcsin x
2 2
3
2
2- -
2,
2
If sin θ = then sine is taking an angle and giving us the ratio of the side opposite to the hypotenuse.
If sin-1 = θ then inverse sine is taking the ratio of the sidesand giving us the angle
Ex2) a) Find the exact value of arcsin(- ½)
b) Find the exact value of arcsin( )
c) Find the exact value of arcsin(2)
r
y
r
y
2
3
Ex3) Graph y = cos x
1
-1 Graph y = arccos x
Ex4) Find the exact value of arccos(½)
Find the exact value of arccos ( )
2 2
3
2
2- -
2
3
Ex5) Graph y = tan x
1
-1
Graph y = tan-1 x
Ex6) Find the approx value of tan-1 (.7042)
Ch4.7A p383 7-19odd (a,b),23,25
2 2
3
2
2- -
Ch4.7A p383 7-19odd (a,b),23,25
Ch4.7A p383 7-19odd (a,b),23,25
Ch4.7B – Inverse Trig Functions cont
Ex6) Graph y = sin
1
-1
Graph y = sin-1
2 2
3
2
2- -
2
x
2
x
Inverse propertiessin(arcsin x) = x arcsin(sin x) = xcos(arccos x) = x arccos(cos x) = xtan(arctan x) = x arctan(tan x) = x
Ex7) Find the exact value of tan(arctan(-5))
Find the exact value of arcsin(sin )3
5
Ch4.7B p383 8-20even (a only),27-41odd
Ch4.7C – Inverse Trig Functions cont
Ex7) Find the exact value of a. tan(arccos( )
b. cos(arcsin( )
5
2
5
3
Ex8) Write each as an algebraic expression: a. sin(arccos(3x) 0 < x < 1/3
b. cot(arccos(3x) 0 < x < 1/3
Ch4.7C p385 34-42even,43-51odd,71,75
Ch4.7C p385 34-42even,43-51odd,71,75
34.2˚
c
b = 19.4
a
Ch4.8A – Applications
Ex1) Solve the triangle: B
CA
72˚
c =1
10ft
h = ?
Ex2) The maximum angle for a ladder is 72˚.If a fire dept’s longest ladder is 110ft, what is the max rescue height?
35˚
h = ?
Ex3) At a point 200ft from the base of a building,the angle of elevation to the bottom of a smoke stack is 35˚.The angle of elevation to the top of the smoke stack is 53˚.Find the height of the smoke stack.
53˚
3.9m
20m
Ex4) Find the angle of depression to the bottom of a pool.
Ch4.8A p394 1-11odd,18
1.3m
Ch4.8A p394 1-11odd,18
Ch4.8B – Applications
Ex5) A ship leaves a port with a heading of N54˚W traveling at 20mph.Ship 2 leaves port at the same tjme with a heading N36˚E traveling at 30mph. After 2 hours how far apart are they?
HW#35) An observer in a lighthouse 350ft above sea level observes2 ships directly offshore. The angle of depression to the ships are 4˚ and 6.5˚. How far apart are the ships?
Ch4.8B p395 17-37odd
4˚ 6.5˚
Lab4.2 – Finding Angles
Ch4.8B p395 17-37odd
Ch4.8B p395 17-37odd
Ch4.8B p395 17-37odd
Ch4.8C – Simple Harmonic Motion (SHM)
10cm
0cm
-10cm
10cm
0cm
-10cm
d = a.sinωt or d = a.cosωt
|a| = amplitude
= period
= frequency
2
2
Ex6) a) Write an equation for the SHM of a ball attached to a spring, that is pushed up 10cm, and oscillates with a period of 4sec.
b) Find the frequency.
Ex7) Given a spring in SHM described by: d = 6cos[ t]find:a) Periodb) Frequencyc) Where is the ball located when t = 4?d) Find 2 times where d = 0.
4
3
Lab4.3 – Simple Harmonic Motion
Go over HW quickly
HW: Finish lab questions+
Ch4 Rev#1 p401 1-53eoeCh4 Rev#2p402 61,65,69,73,75,77, 87,93,97,99,103,104,105,107
Ch4.8C p398 20,49-56all,58 (let’s do 52 and 56 in class #20 on next slide)
Ch4 Rev#1 p401 1-53eoe Ch4 Rev#2p402 61,65,69,73,75,77, 87,93,97,99,103,104,105,107
Ch4 Rev#1 p401 1-53eoe
Ch4 Rev#1 p401 1-53eoe
Ch4 Rev#1 p401 1-53eoe
Ch4 Rev#2p402 61,65,69,73,75,77,87,93,97,99,103,104,105,107
Ch4 Rev#2p402 61,65,69,73,75,77,87,93,97,99,103,104,105,107
654321
-1-2-3-4-5-6
-6 -5 -4 -3 -2-1 1 2 3 4 5 6