Ch4.1A – Radian and Degree Measure
r
Ch4.1A – Radian and Degree Measure
s
θ
~3.14 arcs
= 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 (
¼ revolution (
)=
)=
radians
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
2

1/3 revolution (60˚) =
1/8 revolution (45˚) =
3
radians

radians
4
13
Ex1) Find the acute angle equivalent to
6
Ex2) Find the negative angle equivalent to
3
4
Ex3) Find the positive angle equivalent to 
2
3
Ex4) Find the complement and supplement
angles to
2
a)
5
4
b)
5
Degree/Radian Conversions
Degree/Radian Conversions 90    rad
2
60  

rad
3 
45   rad
4
30   rad
6
0  0 rad
360  2 rad
180   rad

3
270 
rad
2

Conversions:
180    rad


1 
rad
180
Conversion Factor:

 rad
180
or

 rad
180
Ex5) Convert:
a) 135˚
b) -270˚
c) 

2
rad
d) 2 rad
(Quiz on conv in 2 days)
Ch4.1A p318 5-19odd,45-55odd
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:
time
d
v
t
angle swept out
time

omega
Ex3) The second hand of a clock is 10.2cm long.
Find the speed of the second hand.

t
(θ must be in radians)
Ex3) A lawn roller is 30in in diameter and makes 1 revolution
every 5/6 sec.
a) Find the angular speed
b) 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
Convert radians to degrees:
A
B
C
1.  rad
2.
3.
2
3
11
6
1. 2 rad
5
2.
6
5
3.
3
Convert degrees to radians:
A
B
4. 270˚
4. 90˚
5. 60˚
5. 30˚
6. 210˚
6. 240˚
1.
2.

2
Name___________
D
rad

3
1.
rad
2
2.

6
4
3.
3
3
7
3.
6
C
4. 180˚
5. 120˚
6. 330˚
D
4. 360˚
5. 150˚
6. 300˚
Ch4.2 – The Unit Circle
x2 + y2 = 1
Ex1) 45˚ = _____ rad
x = _____
y = _____
Ex2) 60˚ = _____ rad
x = _____
y = _____
Ex3) 30˚ = _____ rad
x = _____
y = _____
Ex4) 0˚ = _____ rad
x = _____
y = _____
Ex5) 90˚ = _____ rad
x = _____
y = _____
Ex6)
x = _____
y = _____
x = _____
y = _____
x = _____
y = _____
x = _____
y = _____
Trig Functions
(sine)
sin t = y
(cosine)
cos t = x
Ex7) Eval 3 trigs for:
a)
t

6
b)
5
t
4
c)
t 
d)
3
t
2
Ch4.2A p328 1-39odd (only sin,cos,tan)
(tangent)
y
tan t =
x
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
1
(cosecant) csc t =
y
1
(secant) sec t =
x
x
(cotangent) cot t =
y
sin t = y
cos t = x
tan t =
y
x
Ex8) Eval 6 trigs for:
t

3
sin t = y
cos t = x
y
tan t =
x
Ex9) Eval:
 13 

 6 
a) sin 
 7 

 2 
b) cos 
1
csc t =
y
1
sec t =
x
x
cot t =
y
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 Trig
Quiz tomorrow on this!
opposite
Θ
adjacent
SOH-CAH-TOA
sin θ =
cos θ =
tan θ =
Ch4.3 – Right Triangle Trig
opposite
SOH-CAH-TOA
Θ
adjacent
opp
sin θ =
hyp
adj
cos θ =
hyp
opp
tan θ =
adj
hyp
csc θ =
opp
hyp
sec θ =
opp
adj
cot θ =
opp
Ex1) Eval 6 trigs for:
5
Θ
3
4
Ch4.3 – Right Triangle Trig
opposite
SOH-CAH-TOA
Θ
adjacent
opp
sin θ =
hyp
adj
cos θ =
hyp
opp
tan θ =
adj
hyp
csc θ =
opp
hyp
sec θ =
opp
adj
cot θ =
opp
Ex2) Find the value of sin45˚, cos45˚, tan45˚
45˚
Ex3) Use the equilateral triangle to find the value of sin60˚, cos60˚,
sin30˚, cos30˚
Sine, Cosine, and Tangent of Special Angles
1
sin30˚ =
2
2
sin45˚ =
2
3
cos30˚ =
2
2
cos45˚ =
2
3
sin60˚ =
2
1
cos60˚ =
2
3
tan30˚ =
3
tan45˚ = 1
tan60˚ =
3
HW#8) Find exact values of 6 trigs for:
3
Θ
6
Ch4.3A p338 1-22(a,b) Quiz tomorrow – would u like 2 c a sample?
Sine, Cosine, and Tangent of Special Angles
1
sin30˚ =
2
2
sin45˚ =
2
3
cos30˚ =
2
2
cos45˚ =
2
3
sin60˚ =
2
1
cos60˚ =
2
3
tan30˚ =
3
tan45˚ = 1
tan60˚ =
3
HW#8) Find exact values of 6 trigs for:
3
Θ
6
Ch4.3A p338 1-22(a,b) Quiz tomorrow – would u like 2 c a sample?
Trigonometry & Vector Components
S in
O pp
H yp
C os
A dj
H yp
T an
O pp
A dj
opp
sinΘ = hyp
adj
cosΘ = hyp
opp
tanΘ = adj
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
Find exact values:˚
A
1. sin 30˚
2. tan 30˚
3. cos 60˚
4. tan 45˚
5. sin 60˚
6. cos 45˚
B
1. cos 30˚
2. tan 60˚
3. sin 60˚
4. cos 45˚
5. cos 60˚
6. tan 45˚
Name___________
C
1. sin 60˚
2. sin 30˚
3. tan 45˚
4. cos 60˚
5. tan 30˚
6. cos 45˚
D
1. cos 60˚
2. cos 30˚
3. tan 30˚
4. sin 60˚
5. tan 45˚
6. sin 45˚
Ch4.3B – Trig Identities
Reciprocals:
1
sin θ =
csc
1
cos θ =
sec
1
tan θ =
cot 
1
csc θ =
sin 
1
sec θ =
cos 
1
cot θ =
tan 
Combos:
sin 
tan  
cos
Pythag:
sin 2   cos2   1
1  tan 2   sec2 
1  cot 2   csc2 
cos
cot  
sin 
Quiz in 2 days on these
identities.
Ex4) Let θ be acute angle, with sin θ = 0.6, find:
a) cos θ
b) tan θ
θ˚
Ex5) Let θ be acute, with tan θ = 3, find:
a) cot θ
b) sec 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:
3
a) sin θ =
2
2
b) cos θ =
2
b) csc θ = 2
Ex8) Use calc to find θ in:
a) degrees for cos θ = 0.3746
b) radians for sin θ = 0.3746
Ch4.3B p339 23-31odd,37-40all(a only),47-55all(a only) Quiz in 2 days
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,81,82
Quiz tomorrow. Would u like to c an example?
Quiz example:
1
=
csc
1
=
tan 
sin 

cos
_____ cos 2   1
1  _____  sec2 
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
Reciprocals
A
B
1.
2.
1

sec
1

cot 
Combinations
3.
sin 

cos
Pythag
4.
1

csc
Name__________
C
D
1

tan 
1

cot 
1

sin 
1

tan 
1

cos
cos

sin 
sin 

cos
cos

sin 
sin 2   ___  1 1  ___  sec2  ___ cos2   1 1  cot2   ___
5. 1  tan
2
2
2
2
1

___

csc

sin


___

1
___

cos
 1
  ___
Ch4.4A – Trig Functions of Any Angle
(x,y)
Ch4.4A – Trig Functions of Any Angle
(x,y)
r
Reminder:
QII
sinθ =
cosθ =
tanθ =
QIII
sinθ =
cosθ =
tanθ =
θ
y
sin θ =
r
x
cos θ =
r
y
tan θ =
x
r
csc θ =
y
r
sec θ =
x
x
cot θ =
y
QI
sinθ =
cosθ =
tanθ =
QIV
sinθ =
cosθ =
tanθ =
Ex1) Let -3,4 be a point on the
terminal side of an angle θ.
find sin,cos,tan.
5
4
Ex2) Given tan θ =  . and cos θ > 0. Find sin θ and sec θ.
Ex3) Evaluate sin and tan at 0,

3
, ,
2
2
Reference Angles
- any given angle has an equivalent angle where
0 > θ > 90˚ , or 0 > θ > π/2.
Ex4) Find ref angle:
a) 300˚
b) 2.3
c) -135˚
Ex5) Evaluate:4
a) cos
3
c) csc
Quiz in 2 days (Ex?)
b) tan (-210˚)
11
4
Ch4.4A p349 2-42eoe (all 6 trigs)
Sample Quiz for Ch4.2/4.3
1.
 7 
sin 

 6 
2.
cos(135  )
3.
sin(135  )
4.
5.
6.
7.
8.
 5 
cos

 6 
 2 
sin 

3


cos(270  )
sin(300  )
cos( )
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
1
Ex6) Let θ be an angle in QII, such that sinθ =
3
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!
Ch4.4B – More Trigs at any Angle
HW#43) Eval sin,cos,tan for:
a) 225˚
#63) Find 2 values in degrees and radians
a) 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
0

6

4

3

2

3
2
2
Ch4.5A – Graphs of Sine and Cosine (Quiz first)
Ex1) What are the values of sine and cosine at:
θ sin θ cos θ
0
0
1

1
2
3
2
6

4

3

2

3
2
2
2
2
2
2
3
2
1
2
1
0
0
-1
-1
0
0
1
Ex2) Graph on a # line: y = sin x
Ex3) Graph on a # line: y = cos x
y = a.sin x
y = a.cos x
Amplitude – stretches and shrinks graph vertically
Ex4) Sketch y = 2.sin x
θ
0

2

3
2
2
2.sinx
y = a.sin x
y = a.cos x
Amplitude – stretches and shrinks graph vertically
Ex5) Sketch y = ½.cos x
θ ½.cos x
0

2

3
2
2
Period of Sine and Cosine
(Normally its 2π)
y = a.sin (bx) y = a.cos (bx)
b determines the period
2
period 
b
Ex6) Sketch y = sin(½x) vs y = sin (x) vs y = sin(2x)
1

–1
Ch4.5A p361 1–13odd, 43,45
2

3
2
2
3
4
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
b determines the period
c = horizontally shifts
the period
2
period 
b
Ex7) Sketch y =
½.sin(x
start: bx – c = 0
end: bx – c = 2π

– )
3

2

3
2
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 Functions
Ex1) Sketch the graph of y = tan x
θ tan θ
0

4

2

3
2
2


4
Ch4.6A – Graphs of Other Functions
Ex1) Sketch the graph of y = tan x
θ tan θ
0

4

2

3
2
2
0
1
undefined
0
undefined
0



2
2
To find asymptotes
for tangent:
bx  c  
bx  c 

2

2
x
Ex2) Sketch the graph of y = tan
2
θ tan θ
0

4

2

3
2
2


4
To find asymptotes
for tangent:
bx  c  
bx  c 

2

2
Ex3) Sketch the graph of y = –3tan2x
θ tan θ
0

4

2

3
2
2


4
To find asymptotes
for tangent:
bx  c  
bx  c 

2

2
Ex4) Sketch the graph of y = cot x
θ cot θ
cot  
0

4

2

3
2
2


4
cos
sin 
x
Ex5) Sketch the graph of y = 2cot
3
θ cot θ
0

4

2

3
2
2


4
Ch4.6A p372 9-12,21,22,24
To find asymptotes
for cotangent:
bx  c  0
bx  c  
Ch4.6A p372 9-12,21,22,24
Ch4.6B – Graphs of Other Functions
Ex6) Sketch the graph of y = csc x
θ sin x csc x
0

4

2

3
2
2
To find asymptotes
for cosecant:
anywhere sin x = 0
Ex7) Sketch the graph of y = sec x
θ cos x sec x
0

4

2

3
2
2
To find asymptotes
for secant:
anywhere cos x = 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
x
2
Ch4.6B p372 13-20all,23,26
Ch4.7 – Inverse Trig Functions
Ex1) Graph y = sin x
1
-
-

2

-1
2

3
2
2
Ch4.7 – Inverse Trig Functions
Ex1) Graph y = sin x
1
-
-

2

-1
2

3
2
2
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 


y
r
If sin θ =
If
sin-1
y
r
=θ
then sine is taking an angle and giving us the ratio
of the side opposite to the hypotenuse.
then inverse sine is taking the ratio of the sides
and giving us the angle
Ex2) a) Find the exact value of arcsin(- ½)
3
b) Find the exact value of arcsin( )
2
c) Find the exact value of arcsin(2)
Ex3) Graph y = cos x
1
-
-

2

-1
2
3
2

Graph y = arccos x
Ex4) Find the exact value of arccos(½)
Find the exact value of arccos (
3
)
2
2
Ex5) Graph y = tan x
1
-
-

2

-1
2

3
2
2
Graph y = tan-1 x
Ex6) Find the approx value of tan-1 (.7042)
Ch4.7A p383 7-19odd (a,b),23,25
Ch4.7A p383 7-19odd (a,b),23,25
Ch4.7A p383 7-19odd (a,b),23,25
Ch4.7B – Inverse Trig Functions cont
x
Ex6) Graph y = sin
2
1
-
Graph y =
sin-1
-
x
2

2

-1
2

3
2
2
Inverse properties
sin(arcsin x) = x
cos(arccos x) = x
tan(arctan x) = x
arcsin(sin x) = x
arccos(cos x) = x
arctan(tan x) = x
Ex7) Find the exact value of tan(arctan(-5))
5
Find the exact value of arcsin(sin
)
3
Ch4.7B p383 8-20even (a only),27-41odd
Ch4.7C – Inverse Trig Functions cont
Ex7) Find the exact value of
a. tan(arccos( 2 )
5
3
b. cos(arcsin(
)
5
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
Ch4.8A – Applications
B
Ex1) Solve the triangle:
a
A
34.2˚
b = 19.4
C
Ex2) The maximum angle for a ladder is 72˚.
If a fire dept’s longest ladder is 110ft,
what is the max rescue height?
h=?
72˚
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.
h=?
53˚
35˚
Ex4) Find the angle of depression to the bottom of a pool.
20m
1.3m
3.9m
Ch4.8A p394 1-11odd,18
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 observes
2 ships directly offshore. The angle of depression to the ships
are 4˚ and 6.5˚. How far apart are the ships?
4˚
Ch4.8B p395 17-37odd
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
|a| = amplitude
2
= period


= frequency
2
or
d = a.cosωt
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.
 3 
6cos[  4 


Ex7) Given a spring in SHM described by: d =
find:
a) Period
b) Frequency
c) Where is the ball located when t = 4?
d) Find 2 times where d = 0.
t]
Lab4.3 – Simple Harmonic Motion
Go over HW quickly
HW: Finish lab questions
+
Ch4 Rev#1 p401 1-53eoe
Ch4 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
6
5
4
3
2
1
-6 -5 -4 -3 -2-1
-1
-2
-3
-4
-5
-6
12 3 4 5 6