Trigonometric
Functions
Chapter 5
Angles and Their
Measure
Section 5.1
Basic Terminology

Ray: A half-line starting at a vertex
V

Angle: Two rays with a common
vertex
Basic Terminology

Initial side and terminal side: The
rays in an angle
Angle shows direction and amount of
rotation
 Lower-case Greek letters denote angles

Basic Terminology



Positive angle: Counterclockwise
rotation
Negative angle: Clockwise rotation
Coterminal angles: Share initial and
terminal sides
Positive angle
Negative angle
Positive angle
Basic Terminology

Standard position:
Vertex at origin
 Initial side is positive x-axis

Basic Terminology

Quadrantal angle: Angle in standard
position that doesn’t lie in any
quadrant
Lies in quadrant II
Quadrantal angle
Lies in quadrant IV
Measuring Angles

Two usual ways of measuring

Degrees


360± in one rotation
Radians

2¼ radians in one rotation
Measuring Angles

Right angle: A quarter revolution

A right angle contains
90±
¼

radians
2

Measuring Angles

Straight angle: A half revolution.

A straight angle contains:
180±
 ¼ radians

Measuring Angles

Negative angles have negative
measure

Multiple revolutions are allowed
Degrees, Minutes and Seconds


One complete revolution = 360±
One minute:
One-sixtieth of a degree
 One minute is denoted 10
 1± = 600


One second:
One-sixtieth of a minute
 One second is denoted 100
 10 = 6000

Degrees, Minutes and Seconds

Example. Convert to a decimal in degrees
Problem: 64±3502700
Answer:

Example. Convert to the D±M0S00 form
Problem: 73.582±
Answer:
Radians

Central angle: An angle whose vertex
is at the center of a circle

Central angles subtend an arc on the
circle
Radians

One radian is the measure of an
angle which subtends an arc with
length equal to the radius of the circle
Radians
IMPORTANT!
Radians are dimensionless
 If an angle appears with no units, it
must be assumed to be in radians

Arc Length

Theorem. [Arc Length]
For a circle of radius r, a central angle of
µ radians subtends an arc whose length s
is
s = rµ
WARNING!

The angle must be given in radians
Arc Length

Example.
Problem: Find the length of the arc of a
circle of radius 5 centimeters subtended
by a central angle of 1.4 radians
Answer:
Radians vs. Degrees

1 revolution = 2¼ radians = 360±



180± = ¼ radians
1± = 1¼8 0 radians
180 ±
1 radian = ¼
Radians vs. Degrees

Example. Convert each angle in
degrees to radians and each angle in
radians to degrees
(a) Problem: 45±
Answer:
(b) Problem: {270±
Answer:
(c) Problem: 2 radians
Answer:
Radians vs. Degrees

Measurements of common angles
Area of a Sector of a Circle

Theorem. [Area of a Sector]
The area A of the sector of a circle of
radius r formed by a central angle of µ
radians is
A =
1
2
r 2µ
Area of a Sector of a Circle

Example.
Problem: Find the area of the sector of a
circle of radius 3 meters formed by an
angle of 45±. Round your answer to two
decimal places.
Answer:
WARNING!

The angle again must be given in
radians
Linear and Angular Speed

Object moving around a circle or
radius r at a constant speed

Linear speed: Distance traveled divided
by elapsed time
v =
s
t
t = time
µ = central angle swept out in time t
s = rµ = arc length = distance traveled
Linear and Angular Speed

Object moving around a circle or
radius r at a constant speed

Angular speed: Angle swept out divided
by elapsed time
!

=
µ
t
Linear and angular speeds are related
v = r!
Linear and Angular Speed

Example. A neighborhood carnival
has a Ferris wheel whose radius is 50
feet. You measure the time it takes
for one revolution to be 90 seconds.
(a) Problem: What is the linear speed (in
feet per second) of this Ferris wheel?
Answer:
(b) Problem: What is the angular speed
(in radians per second)?
Answer:
Key Points








Basic Terminology
Measuring Angles
Degrees, Minutes and Seconds
Radians
Arc Length
Radians vs. Degrees
Area of a Sector of a Circle
Linear and Angular Speed
Trigonometric
Functions: Unit
Circle Approach
Section 5.2
Unit Circle

Unit circle: Circle with radius 1
centered at the origin
Equation: x2 + y2 = 1
 Circumference: 2¼

Unit Circle

Travel t units around circle, starting
from the point (1,0), ending at the
point P = (x, y)

The point P = (x, y) is used to define
the trigonometric functions of t
Trigonometric Functions

Let t be a real number and P = (x, y)
the point on the unit circle
corresponding to t:
Sine function: y-coordinate of P
sin t = y
 Cosine function: x-coordinate of P
cos t = x
 Tangent function: if x  0

Trigonometric Functions

Let t be a real number and P = (x, y)
the point on the unit circle
corresponding to t:

Cosecant function: if y  0

Secant function: if x  0

Cotangent function: if y  0
Exact Values Using Points on
the Circle


A point on the unit circle will satisfy
the equation x2 + y2 = 1
Use this information together with
the definitions of the trigonometric
functions.
Exact Values Using Points on
the Circle

Example. Let t be a real number and
P=
the point on the unit
circle that corresponds to t.
Problem: Find the values of sin t, cos t,
tan t, csc t, sec t and cot t
Answer:
Trigonometric Functions of
Angles


Convert between arc length and
angles on unit circle
Use angle µ to define trigonometric
functions of the angle µ
Exact Values for Quadrantal
Angles

Quadrantal angles correspond to
integer multiples of 90± or of
radians
Exact Values for Quadrantal
Angles

Example. Find the values of the
trigonometric functions of µ
Problem: µ = 0 = 0±
Answer:
Exact Values for Quadrantal
Angles

Example. Find the values of the
trigonometric functions of µ
Problem: µ =
Answer:
= 90±
Exact Values for Quadrantal
Angles

Example. Find the values of the
trigonometric functions of µ
Problem: µ = ¼ = 180±
Answer:
Exact Values for Quadrantal
Angles

Example. Find the values of the
trigonometric functions of µ
Problem: µ =
Answer:
= 270±
Exact Values for Quadrantal
Angles
Exact Values for Quadrantal
Angles

Example. Find the exact values of
(a) Problem: sin({90±)
Answer:
(b) Problem: cos(5¼)
Answer:
Exact Values for Standard
Angles

Example. Find the values of the
trigonometric functions of µ
Problem: µ =
Answer:
= 45±
Exact Values for Standard
Angles

Example. Find the values of the
trigonometric functions of µ
Problem: µ =
Answer:
= 60±
Exact Values for Standard
Angles

Example. Find the values of the
trigonometric functions of µ
Problem: µ =
Answer:
= 30±
Exact Values for Standard
Angles
Exact Values for Standard
Angles

Example. Find the values of the
following expressions
(a) Problem: sin(315±)
Answer:
(b) Problem: cos({120±)
Answer:
(c) Problem:
Answer:
Approximating Values Using a
Calculator
IMPORTANT!


Be sure that your calculator is in the
correct mode.
Use the basic trigonometric facts:
Approximating Values Using a
Calculator

Example. Use a calculator to find the
approximate values of the following.
Express your answers rounded to two
decimal places.
(a) Problem: sin 57±
Answer:
(b) Problem: cot {153±
Answer:
(c) Problem: sec 2
Answer:
Circles of Radius r

Theorem.
For an angle µ in standard position, let
P = (x, y) be the point on the terminal
side of µ that is also on the circle
x2 + y2 = r2. Then
Circles of Radius r

Example.
Problem: Find the exact values of each of
the trigonometric functions of an angle µ
if ({12, {5) is a point on its terminal
side.
Answer:
Key Points







Unit Circle
Trigonometric Functions
Exact Values Using Points on the
Circle
Trigonometric Functions of Angles
Exact Values for Quadrantal Angles
Exact Values for Standard Angles
Approximating Values Using a
Calculator
Key Points (cont.)

Circles of Radius r
Properties of the
Trigonometric
Functions
Section 5.3
Domains of Trigonometric
Functions



Domain of sine and cosine functions is
the set of all real numbers
Domain of tangent and secant
functions is the set of all real
numbers, except odd integer multiples
of = 90±
Domain of cotangent and cosecant
functions is the set of all real
numbers, except integer multiples of
¼ = 180±
Ranges of Trigonometric
Functions

Sine and cosine have range [{1, 1]
{1 · sin µ · 1; jsin µj · 1
 {1 · cos µ · 1; jcos µj · 1


Range of cosecant and secant is
({1, {1] [ [1, 1)
jcsc µj ¸ 1
 jsec µj ¸ 1


Range of tangent and cotangent
functions is the set of all real numbers
Periods of Trigonometric
Functions


Periodic function: A function f with
a positive number p such that
whenever µ is in the domain of f, so is
µ + p, and
f(µ + p) = f(µ)
(Fundamental) period of f: smallest
such number p, if it exists
Periods of Trigonometric
Functions

Periodic Properties:
sin(µ + 2¼) = sin µ
cos(µ + 2¼) = cos µ
tan(µ + ¼) = tan µ
csc(µ + 2¼) = csc µ
sec(µ + 2¼) = sec µ
cot(µ + ¼) = cot µ


Sine, cosine, cosecant and secant have
period 2¼
Tangent and cotangent have period ¼
Periods of Trigonometric
Functions

Example. Find the exact values of
(a) Problem: sin(7¼)
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
Signs of the Trigonometric
Functions

P = (x, y) corresponding to angle µ


Definitions of functions, where defined
Find the signs of the functions
Quadrant
 Quadrant
 Quadrant
 Quadrant

I: x > 0, y > 0
II: x < 0, y > 0
III: x < 0, y < 0
IV: x > 0, y < 0
Signs of the Trigonometric
Functions
Signs of the Trigonometric
Functions

Example:
Problem: If sin µ < 0 and cos µ > 0, name
the quadrant in which the angle µ lies
Answer:
Quotient Identities

P = (x, y) corresponding to angle µ:

Get quotient identities:
Quotient Identities

Example.
Problem: Given
and
, find the exact values of
the four remaining trigonometric
functions of µ using identities.
Answer:
Pythagorean Identities


Unit circle: x2 + y2 = 1
(sin µ)2 + (cos µ)2 = 1
sin2 µ + cos2 µ = 1
tan2 µ + 1 = sec2 µ
1 + cot2 µ = csc2 µ
Pythagorean Identities

Example. Find the exact values of
each expression. Do not use a
calculator
(a) Problem: cos 20± sec 20±
Answer:
(b) Problem: tan2 25± { sec2 25±
Answer:
Pythagorean Identities

Example.
Problem: Given that
µ is in Quadrant II, find cos µ.
Answer:
and that
Even-Odd Properties


A function f is even if f({µ) = f(µ)
for all µ in the domain of f
A function f is odd if f({µ) = {f(µ)
for all µ in the domain of f
Even-Odd Properties

Theorem. [Even-Odd Properties]
sin({µ) = {sin(µ)
cos({µ) = cos(µ)
tan({µ) = {tan(µ)
csc({µ) = {csc(µ)
sec({µ) = sec(µ)
cot({µ) = {cot(µ)


Cosine and secant are even functions
The other functions are odd functions
Even-Odd Properties

Example. Find the exact values of
(a) Problem: sin({30±)
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
Fundamental Trigonometric
Identities

Quotient Identities

Reciprocal Identities

Pythagorean Identities

Even-Odd Identities
Key Points








Domains of Trigonometric Functions
Ranges of Trigonometric Functions
Periods of Trigonometric Functions
Signs of the Trigonometric Functions
Quotient Identities
Pythagorean Identities
Even-Odd Properties
Fundamental Trigonometric Identities
Graphs of the
Sine and Cosine
Functions
Section 5.4
Graphing Trigonometric
Functions


Graph in xy-plane
Write functions as







y
y
y
y
y
y
=
=
=
=
=
=
f(x)
f(x)
f(x)
f(x)
f(x)
f(x)
=
=
=
=
=
=
sin x
cos x
tan x
csc x
sec x
cot x
Variable x is an angle, measured in radians

Can be any real number
Graphing the Sine Function


Periodicity: Only need to graph on
interval [0, 2¼] (One cycle)
Plot points and graph
Properties of the Sine Function

Domain: All real numbers

Range: [{1, 1]

Odd function

Periodic, period 2¼

x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …

y-intercept: 0

Maximum value: y = 1, occurring at

Minimum value: y = {1, occurring at
Transformations of the Graph
of the Sine Functions

Example.
Problem: Use the graph of y = sin x to
graph
Answer:
4
2
3
2
2
-2
-4
2
2
5
2
3
Graphing the Cosine Function


Periodicity: Again, only need to graph
on interval [0, 2¼] (One cycle)
Plot points and graph
Properties of the Cosine
Function








Domain: All real numbers
Range: [{1, 1]
Even function
Periodic, period 2¼
x-intercepts:
y-intercept: 1
Maximum value: y = 1, occurring at
x = …, {2¼, 0, 2¼, 4¼, 6¼, …
Minimum value: y = {1, occurring at
x = …, {¼, ¼, 3¼, 5¼, …
Transformations of the Graph
of the Cosine Functions

Example.
Problem: Use the graph of y = cos x to
graph
Answer:
4
2
3
2
2
-2
-4
2
2
5
2
3
Sinusoidal Graphs



Graphs of sine and cosine functions
appear to be translations of each
other
Graphs are called sinusoidal
Conjecture.
Amplitude and Period of
Sinusoidal Functions

Graphs of functions y = A sin x and
y = A cos x will always satisfy
inequality {jAj · y · jAj

Number jAj is the amplitude
Amplitude and Period of
Sinusoidal Functions

Graphs of functions y = A sin x and
y = A cos x will always satisfy
inequality {jAj · y · jAj
Number jAj is the amplitude

4
4
2
2
3
2
2
2
2
5
2
3
3
2
2
-2
-2
-4
-4
2
2
5
2
3
Amplitude and Period of
Sinusoidal Functions
Period of y = sin(!x) and

y = cos(!x) is
4
4
2
2
3
2
2
2
2
5
2
3
3
2
2
-2
-2
-4
-4
2
2
5
2
3
Amplitude and Period of
Sinusoidal Functions
Cycle: One period of y = sin(!x) or

y = cos(!x)
4
4
2
2
3
2
2
2
2
5
2
3
3
2
2
-2
-2
-4
-4
2
2
5
2
3
Amplitude and Period of
Sinusoidal Functions

Cycle: One period of y = sin(!x) or
y = cos(!x)
Amplitude and Period of
Sinusoidal Functions

Theorem. If ! > 0, the amplitude and
period of y = Asin(!x) and
y = Acos(! x) are given by
Amplitude = j Aj
Period =
.
Amplitude and Period of
Sinusoidal Functions

Example.
Problem: Determine the amplitude and
period of y = {2cos(¼x)
Answer:
Graphing Sinusoidal Functions

One cycle contains four important
subintervals
For y = sin x and y = cos x these are

Gives five key points on graph

Graphing Sinusoidal Functions

Example.
Problem: Graph y = {3cos(2x)
Answer:
4
2
3
2
2
-2
-4
2
2
5
2
3
Finding Equations for
Sinusoidal Graphs

Example.
Problem: Find an equation for the graph.
Answer:
6
4
2
3
5
2
2
3
2
3
2
2
-2
-4
-6
2
2
5
2
3
Key Points







Graphing Trigonometric Functions
Graphing the Sine Function
Properties of the Sine Function
Transformations of the Graph of the
Sine Functions
Graphing the Cosine Function
Properties of the Cosine Function
Transformations of the Graph of the
Cosine Functions
Key Points (cont.)




Sinusoidal Graphs
Amplitude and Period of Sinusoidal
Functions
Graphing Sinusoidal Functions
Finding Equations for Sinusoidal
Graphs
Graphs of the
Tangent, Cotangent,
Cosecant and Secant
Functions
Section 5.5
Graphing the Tangent
Function


Periodicity: Only need to graph on
interval [0, ¼]
Plot points and graph
Properties of the Tangent
Function

Domain: All real numbers, except odd
multiples of

Range: All real numbers

Odd function

Periodic, period ¼

x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …

y-intercept: 0

Asymptotes occur at
Transformations of the Graph
of the Tangent Functions

Example.
Problem: Use the graph of y = tan x to
8
graph
6
4
Answer:
2
3
2
2
-2
-4
-6
-8
2
2
5
2
3
Graphing the Cotangent
Function

Periodicity: Only need to graph on
interval [0, ¼]
Graphing the Cosecant and
Secant Functions


Use reciprocal identities
Graph of y = csc x
Graphing the Cosecant and
Secant Functions


Use reciprocal identities
Graph of y = sec x
Key Points





Graphing the Tangent Function
Properties of the Tangent Function
Transformations of the Graph of the
Tangent Functions
Graphing the Cotangent Function
Graphing the Cosecant and Secant
Functions
Phase Shifts;
Sinusoidal Curve
Fitting
Section 5.6
Graphing Sinusoidal Functions


y = A sin(!x), ! > 0

Amplitude jAj

Period
y = A sin(!x { Á)

Phase shift

Phase shift indicates amount of shift

To right if Á > 0

To left if Á < 0
Graphing Sinusoidal Functions

Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):

Determine amplitude jAj

Determine period

Determine starting point of one cycle:

Determine ending point of one cycle:
Graphing Sinusoidal Functions

Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):

Divide interval
into four
subintervals, each with length

Use endpoints of subintervals to find the
five key points

Fill in one cycle
Graphing Sinusoidal Functions

Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):

Extend the graph in each direction to
make it complete
Graphing Sinusoidal Functions

Example. For the equation
(a) Problem: Find the amplitude
Answer:
(b) Problem: Find the period
Answer:
(c) Problem: Find the phase shift
Answer:
Finding a Sinusoidal Function
from Data

Example. An experiment in a wind tunnel
generates cyclic waves. The following data is
collected for 52 seconds.
Let v represent the wind speed in feet per second
and let x represent the time in seconds.
Time (in seconds), x
Wind speed (in feet per second), v
0
21
12
42
26
67
41
40
52
20
Finding a Sinusoidal Function
from Data

Example. (cont.)
Problem: Write a sine equation that
represents the data
Answer:
Key Points


Graphing Sinusoidal Functions
Finding a Sinusoidal Function from
Data