Trigonometric Functions

1 

Trigonometric 

Functions 

Copyright © 2009 Pearson Addison-Wesley 

1.1-1

1 Trigonometric Functions 

1.1 Angles 

1.2 Angle Relationships and Similar 

Triangles 

1.3 Trigonometric Functions 

1.4 Using the Definitions of the 

Trigonometric Functions 


1.1-2

1.1 

Angles 

Basic Terminology Degree Measure Standard Position 

Coterminal Angles 

Copyright © 2009 Pearson Addison-Wesley 1.1-3 

1.1-3

Basic Terminology 

Two distinct points determine line AB. 

Line segment AB—a portion of the line between A 

and B, including points A and B. 

Ray AB—portion of line AB that starts at A and 

continues through B, and on past B. 


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An angle consists of two 

rays in a plane with a 

common endpoint. 

The two rays are the 

sides of the angle. 

The common endpoint is 

called the vertex of the 

angle. 


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An angle’s measure is 

generated by a rotation 

about the vertex. 

The ray in its initial 

position is called the 

initial side of the angle. 

The ray in its location 

after the rotation is the 

terminal side of the 

angle. 



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Positive angle: The 

rotation of the terminal 

side of an angle is 

counterclockwise. 

Negative angle: The 

rotation of the terminal 

side is clockwise. 


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Types of Angles 

The most common unit for measuring angles is the 

degree. 

A complete rotation of a 

ray gives an angle 

whose measure is 360 . 

of complete rotation gives an angle 

whose measure is 1 . 


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Types of Angles 

Angles are classified by their measures. 


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Example 1 

FINDING THE COMPLEMENT AND THE 

SUPPLEMENT OF AN ANGLE 

For an angle measuring 40 , find the measure of 

its (a) complement and (b) supplement. 

(a) To find the measure of its complement, subtract 

the measure of the angle from 90 . 

Complement of 40 . 

(b) To find the measure of its supplement, subtract 

the measure of the angle from 180 . 

Supplement of 40 . 


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Example 2 

Find the measure of 

each marked angle. 

Since the two angles form 

a right angle, they are 

complementary. 

FINDING MEASURES OF 

COMPLEMENTARY AND 

SUPPLEMENTARY ANGLES 

Combine terms. 

Divide by 9. 

Determine the measure of each angle by substituting 

10 for x: 


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Example 2 

Find the measure of 

each marked angle. 

Since the two angles form 

a straight angle, they are 

supplementary. 


COMPLEMENTARY AND 

SUPPLEMENTARY ANGLES (continued) 

The angle measures are and . 


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Degrees, Minutes, Seconds 

One minute is 1/60 of a degree. 

One second is 1/60 of a minute. 


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Example 3 

CALCULATING WITH DEGREES, 

MINUTES, AND SECONDS 

Perform each calculation. 

(a) 

(b) 

Add degrees 

and minutes 

separately. 

Write 90 as 

89 60′. 


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Example 4 

Find the sum or difference. 

CONVERTING BETWEEN DECIMAL 

DEGREES AND DEGREES, MINUTES, 

AND SECONDS 

(a) 

(b) 

Add degrees 

and minutes 

separately. 

Write 90 as 

89 60′. 


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Standard Position 

An angle is in standard position if its vertex is at 

the origin and its initial side is along the positive 

x-axis. 


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Quandrantal Angles 

Angles in standard position having their 

terminal sides along the x-axis or y-axis, 

such as angles with measures 90 , 180 , 

270 , and so on, are called quadrantal 

angles. 


1.1-17


A complete rotation of a ray results in an angle 

measuring 360 . By continuing the rotation, angles 

of measure larger than 360 can be produced. 

Such angles are called coterminal angles. 

The measures of coterminal angles differ by 360 . 


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Example 5 


COTERMINAL ANGLES 

Find the angle of least possible positive measure 

coterminal with an angle of 908 . 

Add or subtract 360 as 

many times as needed to 

obtain an angle with 

measure greater than 0 but 

less than 360 . 

An angle of 908 is coterminal with an angle of 188 . 


1.1-19

Example 5 


COTERMINAL ANGLES (continued) 


coterminal with an angle of –75 . 

Add or subtract 360 as 

many times as needed to 

obtain an angle with 

measure greater than 0 but 

less than 360 . 

An angle of –75 is coterminal with an angle of 285 . 


1.1-20

Example 5 


COTERMINAL ANGLES (continued) 


coterminal with an angle of –800 . 

The least integer multiple of 

360 greater than 800 is 

An angle of –800 is coterminal with an angle of 280 . 


1.1-21


To find an expression that will generate all angle 

coterminal with a given angle, add integer 

multiples of 360 to the given angle. 

For example, the expression for all angles 

coterminal with 60 is 


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1.1-23

Example 6 

ANALYZING THE REVOLUTIONS OF A 

CD PLAYER 

CAV (Constant Angular Velocity) CD players always 

spin at the same speed. Suppose a CAV player 

makes 480 revolutions per minute. Through how 

many degrees will a point on the edge of a CD move 

in two seconds. 


1.1-24

Example 6 

ANALYZING THE REVOLUTIONS OF A 

CD PLAYER 

Solution 

The player revolves 480 times in one minute or 

times = 8 times per second. 

In two seconds, the player will revolve 

times. 

Each revolution is 360 , so a point on the edge of the 

CD will revolve 

in two seconds. 


1.1-25

1 



Copyright © 2009 Pearson Addison-Wesley 1.2-1


1.1 Angles 


Triangles 





1.2 

Angle Relationships and 

Similar Triangles 

Geometric Properties Triangles 

Copyright © 2009 Pearson Addison-Wesley 1.1-3 1.2-3

Vertical Angles 

Vertical angles have equal measures. 

The pair of angles NMP and 

RMQ are vertical angles. 


Parallel Lines 

Parallel lines are lines that lie in the same plane 

and do not intersect. 

When a line q intersects 

two parallel lines, q, is 

called a transversal. 


Angles and Relationships 

Name 

Alternate interior angles 

Alternate exterior angles 

Interior angles on the same 

side of the transversal 

Corresponding angles 

Angles 

4 and 5 

3 and 6 

1 and 8 

2 and 7 

4 and 6 

3 and 5 

2 & 6, 1 & 5, 

3 & 7, 4 & 8 

Rule 

Angles measures are equal. 

Angle measures are equal. 

Angle measures add to 180 . 

Angle measures are equal. 


Example 1 

FINDING ANGLE MEASURES 

Find the measure of angles 1, 2, 3, and 4, given 

that lines m and n are parallel. 

Angles 1 and 4 are 

alternate exterior angles, 

so they are equal. 

Subtract 3x. 

Add 40. 

Divide by 2. 

Angle 1 has measure 

Substitute 21 

for x. 


Example 1 

FINDING ANGLE MEASURES (continued) 

Angle 4 has measure 

Substitute 21 

for x. 

Angle 2 is the supplement of 

a 65 angle, so it has 

measure . 

Angle 3 is a vertical angle to angle 1, so its measure 

is 65 . 


Angle Sum of a Triangle 

The sum of the measures of the angles of 

any triangle is 180 . 


Example 2 

APPLYING THE ANGLE SUM OF A 

TRIANGLE PROPERTY 

The measures of two of the angles of a 

triangle are 48 and 61 . Find the measure 

of the third angle, x. 

The sum of the 

angles is 180 . 

Add. 

Subtract 109 . 

The third angle of the triangle measures 71 . 


Types of Triangles: Angles 


Types of Triangles: Sides 


Conditions for Similar 

Triangles 

For triangle ABC to be similar to triangle 

DEF, the following conditions must hold. 

1. Corresponding angles must have the 

same measure. 

2. Corresponding sides must be 

proportional. (That is, the ratios of the 

corresponding sides must be equal.) 


Example 3 

FINDING ANGLE MEASURES IN SIMILAR 

TRIANGLES 

In the figure, triangles ABC and NMP are similar. 

Find the measures of angles B and C. 

Since the triangles are similar, corresponding angles 

have the same measure. 

B corresponds to M, so angle B measures 31 . 

C corresponds to P, so angle C measures 104 . 


Example 4 

FINDING SIDE LENGTHS IN SIMILAR 

TRIANGLES 

In the figure, triangles ABC and NMP are similar. 

Find the measures of angles B and C. 

Since the triangles are similar, corresponding sides 

are proportional. 

DF corresponds to AB, and DE corresponds to AC, 

so 


Example 4 

FINDING SIDE LENGTHS IN SIMILAR 

TRIANGLES (continued) 

Side DF has length 12. 

EF corresponds to CB, so 

Side EF has length 16. 


Example 5 

FINDING THE HEIGHT OF A FLAGPOLE 

Firefighters at a station need to measure the height 

of the station flagpole. They find that at the instant 

when the shadow of the station is 18 m long, the 

shadow of the flagpole is 99 ft long. The station is 

10 m high. Find the height of the flagpole. 

Since the two triangles 

are similar, corresponding 

sides are proportional. 


Example 5 

FINDING THE HEIGHT OF A FLAGPOLE 

(continued) 

Lowest terms 

The flagpole is 55 feet high. 


1 





1.1 Angles 


Triangles 





1.3 



Quadrantal Angles 



Let (x, y) be a point other the origin on the terminal 

side of an angle in standard position. The 

distance from the point to the origin is 



The six trigonometric functions of θ are 

defined as follows: 


Example 1 

FINDING FUNCTION VALUES OF AN 

ANGLE 

The terminal side of angle in standard position 

passes through the point (8, 15). Find the values of 

the six trigonometric functions of angle . 


Example 1 


ANGLE (continued) 


Example 2 


ANGLE 

The terminal side of angle in standard position 

passes through the point (–3, –4). Find the values 

of the six trigonometric functions of angle . 


Example 2 



Use the definitions of the trigonometric functions. 


Example 3 


ANGLE 

Find the six trigonometric function values of the 

angle θ in standard position, if the terminal side of θ 

is defined by x + 2y = 0, x ≥ 0. 

We can use any point on 

the terminal side of to 

find the trigonometric 

function values. 

Choose x = 2. 


Example 3 



The point (2, –1) lies on the terminal side, and the 

corresponding value of r is 

Multiply by 

the denominators. 

to rationalize 


Example 4(a) 

FINDING FUNCTION VALUES OF 

QUADRANTAL ANGLES 

Find the values of the six trigonometric functions for 

an angle of 90 . 

The terminal side passes 

through (0, 1). So x = 0, y = 1, 

and r = 1. 

undefined 

undefined 


Example 4(b) 

FINDING FUNCTION VALUES OF 

QUADRANTAL ANGLES 

Find the values of the six 

trigonometric functions for an 

angle θ in standard position 

with terminal side through 

(–3, 0). 

x = –3, y = 0, and r = 3. 

undefined 

undefined 


Undefined Function Values 

If the terminal side of a quadrantal angle lies along 

the y-axis, then the tangent and secant functions 

are undefined. 

If the terminal side of a quadrantal angle lies along 

the x-axis, then the cotangent and cosecant 

functions are undefined. 


Commonly Used Function 

Values 

 

sin 

cos 

tan 

cot 

sec 

csc 

0 

0 

1 

0 

undefined 

1 

undefined 

90 

1 

0 

undefined 

0 

undefined 

1 

180 

0 

1 

0 

undefined 

1 

undefined 

270 

1 

0 

undefined 

0 

undefined 

1 

360 

0 

1 

0 

undefined 

1 

undefined 


Using a Calculator 

A calculator is degree mode 

returns the correct values 

for sin 90 and cos 90 . 

The second screen shows 

an ERROR message for tan 

90 because 90 is not in 

the domain of the tangent 

function. 


Caution 

One of the most common errors 

involving calculators in trigonometry 

occurs when the calculator is set for 

radian measure, rather than degree 

measure. 


1 





1.1 Angles 


Triangles 





1.4 

Using the Definitions of the 


Reciprocal Identities Signs and Ranges of Function Values 

Pythagorean Identities Quotient Identities 


Reciprocal Identities 

For all angles θ for which both functions are 

defined, 


Example 1(a) 

USING THE RECIPROCAL IDENTITIES 

Since cos θ is the reciprocal of sec θ, 


Example 1(b) 

USING THE RECIPROCAL IDENTITIES 

Since sin θ is the reciprocal of csc θ, 

Rationalize the 

denominator. 


Signs of Function Values 

in 

Quadrant 

sin 

cos 

tan 

cot 

sec 

csc 

I 

+ 

+ 

+ 

+ 

+ 

+ 

II 

+ 

 

 

 

 

+ 

III 

 

 

+ 

+ 

 

 

IV 

 

+ 

 

 

+ 

 


Ranges of Function Values 


Example 2 

DETERMINING SIGNS OF FUNCTIONS 

OF NONQUADRANTAL ANGLES 

Determine the signs of the trigonometric functions of 

an angle in standard position with the given measure. 

(a) 87 

The angle lies in quadrant I, so all of its 

trigonometric function values are positive. 

(b) 300 

The angle lies in quadrant IV, so the cosine and 

secant are positive, while the sine, cosecant, 

tangent, and cotangent are negative. 


Example 2 

DETERMINING SIGNS OF FUNCTIONS 

OF NONQUADRANTAL ANGLES (cont.) 

Determine the signs of the trigonometric functions of 

an angle in standard position with the given measure. 

(c) –200 

The angle lies in quadrant II, so the sine and 

cosecant are positive, while the cosine, secant, 

tangent, and cotangent are negative. 


Example 3 

IDENTIFYING THE QUADRANT OF AN 

ANGLE 

Identify the quadrant (or quadrants) of any angle 

that satisfies the given conditions. 

(a) sin > 0, tan < 0. 

Since sin > 0 in quadrants I and II, and tan < 0 in 

quadrants II and IV, both conditions are met only in 

quadrant II. 

(b) cos > 0, sec < 0 

The cosine and secant functions are both negative 

in quadrants II and III, so could be in either of 

these two quadrants. 


Ranges of Trigonometric 



Example 4 

DECIDING WHETHER A VALUE IS IN 

THE RANGE OF A TRIGONOMETRIC 

FUNCTION 

Decide whether each statement is possible or 

impossible. 

(a) sin θ = 2.5 

Impossible 

(b) tan θ = 110.47 

Possible 

(c) sec θ = .6 

Impossible 


Example 5 

FINDING ALL FUNCTION VALUES 

GIVEN ONE VALUE AND THE 

QUADRANT 

Suppose that angle is in quadrant II and 

Find the values of the other five trigonometric 

functions. 

Choose any point on the terminal side of angle . 

Let r = 3. Then y = 2. 

Since is in quadrant II, 


Example 5 



QUADRANT (continued) 

Remember to 

rationalize the 

denominator. 


Example 5 





Pythagorean Identities 

For all angles θ for which the function values are 

defined, 


Quotient Identities 

For all angles θ for which the denominators are 

not zero, 


Example 6 

FINDING OTHER FUNCTION VALUES 


QUADRANT 

Choose the positive 

square root since sin θ >0. 


Example 6 




To find tan θ, use the quotient identity 


Caution 

In problems like those in Examples 5 

and 6, be careful to choose the 

correct sign when square roots are 

taken. 


Example 7 



QUADRANT 

Find sin θ and cos θ, given that 

quadrant III. 

and θ is in 

Since θ is in quadrant III, both sin θ and cos θ are 

negative. 


Example 7 




Caution 

It is incorrect to say that sin θ = –4 

and cos θ = –3, since both sin θ and 

cos θ must be in the interval [–1, 1]. 


Example 7 




Use the identity 

to find sec θ. Then 

use the reciprocal identity to find cos θ. 

Choose the negative 

square root since sec θ

Example 7 




Choose the negative 

square root since sin θ

Example 7 




This example can also be 

worked by drawing θ is 

standard position in quadrant 

III, finding r to be 5, and then 

using the definitions of sin θ 

and cos θ in terms of x, y, 

and r.

Trigonometric Functions

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