Math 100
Section 6.1
Angle Measure
Definitions:
An angle consists of two rays, R1 and R2, with a common vertex. Often we think of an angle
as being a rotation of R1 onto R2. R1 is called the initial side and R2 is called the terminal
side. When the rotation is counterclockwise, the angle is said to be positive, and when the
rotation is clockwise, the angle is negative.
Exercise 1 Label the initial side, terminal side and vertex of the following angles.
Identify each angle as being positive or negative.
Definition:
Angle is said to be in standard position if its initial side lies along the positive x-axis an its
vertex is at the origin. Label the initial side, terminal side and vertex of the angle below.
y
x
Math 100 Chapter 6
117
The measure of an angle is how much rotation is required to move R1 onto R2. Angles
can be measured in degrees. You are probably familiar with degrees: 360 degrees
is one complete revolution, 90 degrees is a right angle, and so on. Another unit used
for measuring angles is radians. One radian is the angle subtended by a circular arc
the length of the circle’s radius:
360 degrees = one complete revolution = 2 radians
Exercise 2 Determine how many degrees are in one radian. Sketch a angle of
about one radian in standard position on the circle below.
1 radian = _______ degrees
Exercise 3 Converting between radians and degrees
a) Express 90 degrees in radians
b) Express /3 radians in degrees
Math 100 Chapter 6
118
Exercise 4 A circle divided up into equal sized segments is shown. Fill in the blanks
below, giving radian measures as multiples of  and reduce fractions to lowest
terms.
Angle of one revolution (once around the
circle) is
________ degrees =_______ radians
Number of segments = ______
Angle within each segment is
_______ degrees = ________ radians
Exercise 5 A circle is divided up into equal sized segments is shown. Fill in the
blanks below, giving radian measures as multiples of  and reducing fractions to
lowest terms.
Angle of one revolution (once around the
circle) is
________ degrees =_______ radians
Number of segments = ______
Angle within each segment is
_______ degrees = ________ radians
Math 100 Chapter 6
119
Exercise 6 Angles in standard position are shown. Give the angle measure of each
in both degrees and radians.
____ degrees = ____radians
____ degrees = ____radians
____ degrees = ____radians
____ degrees = ____radians
____ degrees = ____radians
____ degrees = ____radians
radians
____ degrees = ____radians
Math 100 Chapter 6
____ degrees = ____radians
120
_____ degrees = _____ radians
_____ degrees = _____ radians
_____ degrees = _____ radians
_____ degrees = _____ radians
____ degrees = ____ radians
____ degrees = ____ radians
____ degrees = ____ radians
____ degrees = ____ radians
Math 100 Chapter 6
121
____ degrees = ____ radians
____ degrees = ____ radians
____ degrees = ____ radians
____ degrees = ____ radians
Coterminal Angles
Two angles in standard position are said to be coterminal if they have the same
initial and terminal sides. Coterminal angles look the same after they are made, but
if you watched the angles being swept out, they might appear to be different. For
example, one might be positive and one negative, or one might "wrap around" more
times then another.
Exercise 7 Find two angles that are coterminal with the given angle. Draw the
coterminal angles in standard position.
Math 100 Chapter 6
122
Exercise 8 For each angle:
 convert from radian to degrees—or degrees to radians
 draw in standard position
 state two other angles that are coterminal with the given angle

7
6
radians = _____ degrees
Two coterminal angles are:
____________
 
5
4
____________
radians = _____ degrees
Two coterminal angles are:
____________
____________
   43 radians = _____ degrees
Two coterminal angles are:
____________
 
3
2
____________
radians = _____ degrees
Two coterminal angles are:
____________
Math 100 Chapter 6
____________
123
  60 degrees = _____ radians
Two coterminal angles are:
____________
____________
  120 degrees = _____ radians
Two coterminal angles are:
____________
____________
  1080 degrees = _____ radians
Two coterminal angles are:
____________
____________
  540 degrees = _____ radians
Two coterminal angles are:
____________
Math 100 Chapter 6
____________
124
Objectives and Suggested Exercies Section 6.1
Suggested Exercises are on pages 413 - 414
Objectives
1) To convert angles between radian and degree measure (# 1 – 17 odd)
2) To draw an angle in standard position.
3) To understand the concept of coterminal angles (# 19 – 41 odd)
Math 100 Chapter 6
125
Math 100
Section 6.2
Trigonometry of Right Triangles
sin() 
opposite
hypotenuse
csc() 
Side opposite 
Trigonometric Function Values of an Acute Angle 
Let  be an acute angle of a right triangle. Then the six
trigonometric functions of  are as follows:
hypotenuse
hypotenuse
opposite

adjacent
cos() 
hypotenuse
tan() 
opposite
adjacent
hypotenuse
sec() 
adjacent
cot( ) 
Side adjacent to 
adjacent
opposite
Mnemonics for remembering the trigonometric ratios:
SOHCAHTOA
Sine:
Opposite over
Hypotenuse;
Cosine:
Adjacent over
Hypotenuse;
Tangent:
Opposite over
Adjacent
Oh, Heck, Another Hour Of Algegra. . .
Opposite over
Hypotenuse;
Adjacent over
Hypotenuse;
Opposite over
Adjacent
with
Some Crazy Teacher!
I
o
a
n
s
n
Exercise 1 Find the exact value of the following trigonometric expressions for the
given right triangle:
Sin() = ___________ Sin() = ___________

Cos() = ___________ Cos() = ___________
3
Tan() = ___________ Tan() = ___________
Math 100 Chapter 6

7
126
Special Triangles
Two types of right triangles, 45-45-90 and 30-60-90, have trigonometric
ratios that can be calculated exactly using geometry. These triangles were popular
as examples in the days before calculators, and they still commonly appear in
trigonometry and calculus textbooks.
Exercise 2 45 (/4 radians) Right Triangle
This triangle is obtained by drawing a diagonal in a square of side 1 (shown below).
In the diagram to the right:
 Label the length of the
sides, and the angles in
degrees and radians.
 Use the Pythagorean
Theorem to find the length
of the diagonal
Use the above results to find the exact value of each of the following. Verify your
results by finding a decimal approximation using the trigonometric functions on your
calculator:
Sin ( 45 ) = Sin ( /4 ) = ___________________
Cos (45 ) = Cos (/4 ) = ___________________
Tan (45 ) = Tan ( /4 ) = __________________
Sec ( 45 ) = Sec( /4 ) = ___________________
Csc ( 45) = Csc( /4 ) = ___________________
Cot ( 45 ) = Cot ( /4 ) = __________________
Math 100 Chapter 6
127
Exercise 3 30/60 Right Triangle (/6, /3 radians)
This triangle is obtained by drawing a perpendicular bisector in an equilateral
triangle of side 2 (shown below).
In the diagram to the right:
 Label the length of the
sides, and the angles in
degrees and radians.
 Use the Pythagorean
Theorem to find the length
of the perpendicular
bisector
Use the above results to find the exact value of each of the following. Verify your
results by finding a decimal approximation using the trigonometric functions on your
calculator:
Sin( 30 ) = _________________
Sin ( 60 ) = ___________________
Cos ( /6 ) = ________________ Cos (/3 ) = ____________________
Tan ( 30 ) = ________________ Tan ( /3 ) = ____________________
Math 100 Chapter 6
128
Exercise 4
Evaluate the following expressions exactly:
a) sin(30)cos(30) + sin(60)cos(60)
b) sin2(/3) + cos2(/3)
c) 2sin(/4)cos(/4)
d)
1  sin 2 (  / 6)
Math 100 Chapter 6
129
Exercise 5 In each triangle, an angle and the length of one side is given. Use
trigonometric functions to find the value of the unknown side x. (Not drawn to
scale)
22
x
x
17
10 cm
12 cm
In the previous exercise the angle was given and we used the angle to find the sine,
cosine or tangent.
Angle:
 = 22
Tangent
of 22 
Tan()
______
In the following problems, we are given the sine, cosine or tangent, and we want to
find the angle. The inverse trigonometric functions , pronounced as arcsin(x),
arcos(x), and arctan(x), but often denoted as sin-1(x), cos-1(x), and tan-1(x), serve
this purpose.
Exercise 6 Find the angle whose tangent is .5774
Angle:

Tan-1()
Tangent = .5774
_____
Math 100 Chapter 6
130
Exercise 7 Use inverse trig functions on your calculator to find the unknown angle
 in the following triangles
12 cm
14cm
17cm


10 cm
Exercise 8 Sketch a triangle that has acute angle , and find the other five
trigonometric ratios of . Sin  = ¾
cos = _____
tan = _____
csc = _____
sec = _____
cot = _____
Use the arctan(x) , arcsin(x), or arcos(x) function to find  in:
degrees: ________
Math 100 Chapter 6
radians:________
131
Objectives and Suggested Exercises Section 6.2
Suggested Exercises are on pages 422 - 425
Objectives
1) To find the values of sinx, cosx, tanx, cscx, and secx when x is an acute angle in
a right triangle (# 1, 3, 5)
2) To learn about the “special” angles: 45, 30 and 60 degrees (# 7, 9, 21, 23, 25)
3) To use the trig functions on your calculator to find the length of an unknown
side in a triangle (#7, 9)
4) To find an unknown angle using the inverse trig functions on your calculator.
Math 100 Chapter 6
132
Math 100
Section 6.3
Trigonometric Functions of Angles
This section combines the ideas of section 6.1 (angle measure) and 6.2 (righttriangle trigonometry). In section 6.2, we learned how to find the trigonometric
functions of angles in right-triangles (recall: SOHCAHTOA). In this section we
learn how to find the trigonometric function of any angle in standard position, even
those that are not in the range 0<  < 90 and therefore can’t be in a right
triangle.
Exercise 1 A right triangle with acute angle  and legs of length 3 and 4 is shown. Redraw
the triangle so that  is in standard position--with the side adjacent to  along the positive
x-axis and the right-angle away from the origin.
4
P

Opposite = 3
Hypotenuse = r
3
2
1
Adjacent=4
1
2
3
4
5
a) after the triangle is redrawn answer the following:
i)
what is the x-coordinate of point P? x = ______
ii)
what is the y-coordinate of point P? y = ______
b) what is the length of the hypotenuse? r = ______
c) find the value of the following:
i) sin  = ________
ii) cos  = _______
iii) tan  = _______
iv) csc  = _______
v) sec  = ________
vi) cot  = ________
Math 100 Chapter 6
133
Exercise 2 Label the hypotenuse, the side opposite , and the side adjacent to  on
the right-triangle ABC. Redraw the triangle with the side adjacent to  along the
positive x-axis and the right-angle away from the origin.
y
B
A

x
C
a) after the triangle is redrawn, label the coordinates of point B as (x,y)
Answer the rest of the questions in terms of x and y:
a) what is the length of the side opposite ?________
b) what is the length of the side adjacent to ?________
b) what is the length of the hypotenuse, r?__________
d) find the following:
i) sin  = ______
ii) cos  = ______
iv) csc  = _____
v) sec  = ______
Math 100 Chapter 6
iii) tan  = ______
vi) cot  = ______
134
The idea of the previous
exercise is extended to
angles that do not have a
terminal arm in the first
quadrant. Look at the
definitions shown in the
box and compare them to
the results of the last
exercise.
Let  be an angle in standard position and let P(x,y) be a point on
the terminal side. If r  x 2  y 2 is the distance from the origin
to the point P(x,y), then:
sin( ) 
y
x
r
r
sec( ) 
x
cos( ) 
r
r
csc() 
y
tan( ) 
y
x
x
cot( ) 
y
Exercise 3 Using the above definitions, find the values (where possible) of the
indicated trigonometric functions for the angles shown. In each case, pick one
point (call it P) on the terminal arm of the angle. Is it important which point you
use?
1.
sin  = _______
P( ____ , ____ )
cos  = _______
x = ____
tan  = _______
r = _______________
2.
sin  = _______
P( ____ , ____ )
cos  = _______
x = ____
tan  = _______
r = _______________
3.
sin  = _______
P( ____ , ____ )
cos  = _______
x = ____
tan  = _______
Math 100 Chapter 6
y = ____
y = ____
y = ____
r = _______________
135
Whether a trigonometric function is positive or negative depends only on what
quadrant the terminal arm lies in.
Exercise 4 Fill in the table below, by referring to the coordinate system shown.
Assume (x,y) is a point on the terminal arm of angle :
Terminal
arm in
quadrant
The Sign (+ or -) of :
x
y r
Sin  =y/r
Cos  = x/r
II
Tan  = y/x
I
II
III
IV
III
These results are summarized in the coordinate system: 
You can remember this as “all students take calculus”
Or “all schools torture children”
Or “are simpletons teaching courses?”
Sin +
Tan +
I
IV
All +
Cos +
Reference Angles:
Let  be an angle in standard position. The reference
angle  associated with  is the acute angle formed by
the terminal side of  and the x-axis.
Exercise 5
Sketch the angle
 = 5/3, find
the reference
angle  = _____
Math 100 Chapter 6
Sketch the angle
 = 3/4 and find
the reference
angle
 = ______
136
The reference angle can be used in conjunction with knowledge about which
quadrant the angle lies in to find the exact value of the trigonometric functions.
To find the exact value of trigonometric functions for any angle , carry out the
following steps:
1. Find the reference angle  associated with the angle .
2. Determine the sign (+ or - ) of the trigonometric function with the angle .
3. The value of the trigonometric function of  is the same, except possibly for
sign , as the value of the trigonometric functions of  .
Exercise 6
Find the exact value of the following trigonometric functions:
a) cos (5/6)
reference angle =  = ______
b) sin (3/4)
reference angle =  = ________
c) tan (-/4)
Math 100 Chapter 6
cos(  ) = _________
reference angle =  = ________
sin(  ) = _________
tan(  ) = ________
137
Exercise 7
 Find the value of sin() and cos() from the information given.
 Use the arctan(x) function on your calculator (tan-1x ), in conjunction with
knowledge about what quadrant the terminal arm lies in, to find the approximate
value of  in degrees. In each case, assume 0 <  < 2
a) tan  = 3/4, sin() > 0
sin  = _______
cos  = _______
  _______
b) tan  = 3/4, sin() < 0
sin  = _______
cos  = _______
  _______
c) tan  = -3/4, cos  < 0
sin  = _______
cos  = _______
  _______
Math 100 Chapter 6
138
Objectives and Suggested Exercises Section 6.3
Suggested Exercises are on pages 433 - 434
Objectives
1) To find the reference angle for a given angle. (# 1, 3, 5)
2) To find the values of trigonometric functions for angles that are not
(necessarily) acute. (#7 – 29 odd)
3) To determine the quadrant in which an angle lies (#31, 33)
4) To find the values of trigonometric functions of an angle, given the value of one
of the functions, and one other piece of information. (# 41 – 47 odd)
Math 100 Chapter 6
139
Math 100
Section 5.3
Graphs of Trigonometric Functions
Exercise 1 On an unusual new amusement park ride, riders are submerged in water
(enclosed inside a water-tight car) for a portion of the ride. The ride is similar to a Ferris
wheel with its bottom half underwater, as shown. The duration of the ride is 360 seconds
(6 minutes ). You board the ride at water level (at the point indicated in the drawing).
During the ride you are rotated counterclockwise for one complete revolution. The radius r
of the ride is 1 dekametre (10 metres).
You ride in a water-tight car
a) How many degrees do you rotate in
one second?_________
You rotate counterclockwise
b) Mark your location after: 45, 90, 180,
You start here
270 and 360 seconds
c) During what time intervals will be
water level
going up?
d) During what time interval will you be
going down?
e) What is the maximum elevation above the water that you will reach?
f) What is the greatest depth below the water that you will reach?
g)
You reach the greatest depth __________ after the ride begins.
h) During what time interval will you be above the water?
i)
During what time interval will you be below the water?
Math 100 Chapter 6
140
In the following questions, let t be the time that has
elapsed since the ride began. Recall that the radius of the
ride is 1 dekametre.
a) At what times t will you reach the point labeled A, B, C
and D ?
A
B
C
A_______
B_______
water level
C_______ D________
D
b) Use trigonometry to estimate your elevation (in dekametres) from water-level (to 3
decimal places) at the points A, B and C. In each case, state the time t and the angle 
you have rotated to arrive at the point. (When you are above the water, your elevation
is positive. When you are below the water, your elevation is negative.)
i)
A
t =  = _________
C
A
ii)
B
iii)
C
iv)
D
B
t =  = __________
t =  = ___________
Math 100 Chapter 6
t =  = ___________
More generally, express your elevation as a
function of t:
f( t ) = ____________
141
Exercise 2: The sine function graph
t
Elevation = sin(t)
1.
Complete the table, finding your elevation at 30
second intervals during the ride.
2. Plot the points in the table on the graph below.
0
30
60
90
120
3. Draw a smooth curve through the points. You
have sketched a graph of y = sin(t).
For what values of t is:
sin(t) increasing? (when are you going up?)
sin(t) decreasing? (when are you going down?)
150
180
210
240
sin(t) < 0? (when are you under water?)
sin(t) > 0? (when are you above water?)
sin(t) = 0? (when are you at water level?)
270
300
What is the maximum value of sin(t)? (what is your
highest elevation?)
330
360
What is the minimum value of sin(t)? (what is your
lowest elevation?)
water
level
What would the graph of your elevation at time t look like if you stayed on the ride for more than one
revolution?
Math 100 Chapter 6
142
Transformations of sin(x)
Exercise 3: The graph of y = sin(x) for x from -720 to 720 (-4 to 4 radians) is given.
Plot the following transformations of sin(x). In each case state the amplitute:
1. y = sin(x) + 1
amplitude:________
2. y = sin(x + 90)
3. y = 2sin(x)
4. y = -2.5sin(x)
Math 100 Chapter 6
(in radians, y = sin(x + /2)
amplitude________________
amplitude________________
amplitude________________
143
The Graph of the Cosine Function
In the last exercise, we graphed the function f(x) = sin(x). In this exercise, we
graph the cosine function. The graph of the cosine looks like the graph of the sine,
only it is shifted horizontally.
Exercise 4:
Angle in
Angle in
Cos()
radians ()
degrees ()
0
0
/6
30
/3
60
/2
90
2/3
120
5/6
150

180
7/6
210
4/3
240
3/2
270
5/3
300
11/6
330
2
360
cos()
Math 100 Chapter 6
144
The Period of a Sine Curve
The period of a sine curve is the length of time it takes to complete one complete
revolution. The period of the standard sine curve f(x) = sin(x) is 360 or 2 radians. If k is
a constant, the period of sin(kx) is 360/k or 2/k.
You ride in a sealed car
Exercise 5 Recall the amusement park ride of
Exercise 1. Suppose the ride rotates 2 degrees
every second (instead of 1 degree every second,
as in the original question. Complete the table
and answer the questions below:
a) How long will it to reach the highest point, 1
dam above the water?_________
b) How long will it take to reach the lowest
point?________
c) During what time interval are you going
down?_________
d) If the ride lasts 360 seconds, how many times
will you go around?_____
Time
Degrees
rotated
0
30
60
90
120
Elevation
Rotate counter-clockwise
clocounterclockwise
You start here
water level
The graph of sin(x) is given in gray below. Recall that
sin(x) represents your elevation at time t when the
ride rotates 1 degree per second. Complete the table
at the left and draw a graph of your elevation on the
ride when it rotates at 2 degrees per second.
You graphed the function___________
It has period __________________
150
180
210
240
270
300
330
360
Math 100 Chapter 6
145
Exercise 6 Recall again the amusement park ride of Exercise 1. Suppose the ride rotates
1/2 degrees every second (instead of 1 degree every second, as in the original question.
Complete the table and answer the questions
You ride in a sealed car
below:
a) How long will it to reach the highest point,
1 dam above the water?_________
b) How long will it take to reach the lowest
point?________
c) During what time interval are you going
down?_________
d) If the ride lasts 720 seconds, how many
times will you go around?_____
Time
Degrees
rotated
rotate counterclockwise
You start here
water level
Elevation
0
60
120
180
240
300
360
420
480
540
600
660
The graph of sin(x) is given in gray below. Recall that
sin(x) represents your elevation at time t when the
ride rotates 1 degree per second.
How many times would you go around in 720 seconds on
the ride at 1 degree per second?________
Complete the table at the left and draw a graph of
your elevation on the ride when it rotates at 1/2
degrees per second.
You graphed the function___________
It has period __________________
720
Math 100 Chapter 6
146
Exercise 7 Graph each of the following functions and state the period.
a) f(x) = cos(2x) (The graph of cos(x) is given in gray.)
period = _________________
b) g(x) = cos(x/2)
(The graph of cos(x) is given in grey.)
period = _________________
c) h(x) =
3
2
sin(4x)
(The graph of sin(x) is given in gray.)
period = _________________ amplitude_________________
Math 100 Chapter 6
147
Objectives and Suggested Exercises from Section 5.3
Suggested Exercises are on page 380
Objectives:
1) To be able to sketch the graphs of the sine and cosine functions (page 370)
2) To be able to sketch the graphs of transformations of the sine and cosine
functions, in particular:
a) Vertical shifts (#1)
b) Reflections (# 2)
c) Vertical stretching (change in amplitude) (# 5)
d) Horizontal stretching (change in period) (#11, 13)
e) Horizontal shifts (#19, 21)
3) To determine the amplitude, period and horizontal shift from the graph of a
sine or cosine curve, and write an equations that represents the curve (#33, 35)
Math 100 Chapter 6
148
Math 100
Section 7.1
Trigonometric Identities
Definitions of Trigonometric Functions
(recall from section 6.3)
P(x,y) is a point on the terminal arm and r  x 2  y 2
sin() 
y
x
r
r
sec() 
x
cos() 
r
r
csc() 
y
y
x
x
cot() 
y
Pythagorean Identities
Basic Identities
1
csc() =
sin()
tan() 
1
sec() =
cos()
sin2 ()  cos2 ()  1
tan 2 ()  1  sec 2 ()
cot() =
1
tan( )
cot() =
cos()
sin()
tan() =
sin()
cos()
1  cot 2 ()  csc2 ()
Exercise 1 Use the trigonometric definitions to verify that the basic
identities are true
a) csc() =
1
sin()
b) sec() =
1
cos()
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c) cot() =
1
tan( )
d) tan() =
sin()
cos()
e) cot() =
cos()
sin()
Exercise 2 Use the definitions to show that the first Pythagorean identity is true:
sin2 ()  cos2 ()  1
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Exercise 3 Use the basic identities (not the definitions) to write each expression
in terms of sines and/or cosines, and then simplify
a)
sec(x )
tan( x )
b)
cot( x )
csc(x )
c)
sin( x ) cos(x )

csc(x ) sec(x )
d)
1
1

2
sin ( x ) tan 2 ( x )
e) (1 - sin(x))(1 + sin(x))
g)
(cos(x) tan( x)  1)(sin(x)  1)
cos2 ( x)
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Guidelines for Proving Trigonometric Identites
 Pick one side (usually the more complicated side) and use
algebra and the basic identities to transform it into the other
side, OR, transform both sides into a common expression
 Always keep the two sides separate and independent.
 Never, for example, add or multiply both sides by the same
thing. Remember that proving an identity is not the same as
solving an equation.
 Frequently it is useful to begin by putting both sides in terms
of sine and/or cosine, and simplifying.
Exercise 4 Prove that each of the following equations is an identity:
a) tan( x) cos(x )  csc(x) sin2 ( x )  2 sin( x)
b) (1 + sin(x))2 + cos2(x) = 2 + 2sin(x)
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c) tan(x) + cot(x) = sec(x)csc(x)
d)
csc(x ) cot( x ) tan( x )


cot( x ) csc(x ) csc(x )
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e)
1  sin2 ( x ) csc(x )  1

1  sin( x )
csc(x )
f)
sin(x )
csc(x )  1

sin(x )  1
cot 2 ( x )
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Objectives and Suggested Exercises for Section 7.1
Suggested Exercises are on pages 466 - 467
Objectives:
1) To learn and understand the basic trigonometric identities. (page 461)
2) To write a trigonometric expression in terms of sine and cosine (# 1, 3, 5)
3) To simplify a trigonometric expression using the basic identities (# 7 – 19 odd)
4) To verify various trigonometric identities. (do a selection from # 21 – 81 odd)
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