1 Trigonometric Functions © 2008 Pearson Addison-Wesley. All rights reserved.

1

Trigonometric Functions

© 2008 Pearson Addison-Wesley.All rights reserved

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 1-2

1.1 Angles

1.2 Angle Relationships and Similar Triangles

1.3 Trigonometric Functions

1.4 Using the Definitions of the Trigonometric Functions

Trigonometric Functions1


Angles1.1Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles


For an angle measuring 55°, find the measure of its complement and its supplement.

1.1 Example 1 Finding the Complement and the Supplement of an Angle (page 3)

Complement: 90° − 55° = 35°

Supplement: 180° − 55° = 125°


Find the measure of each angle.

1.1 Example 2(a) Finding Measures of Complementary and Supplementary Angles (page 3)

The two angles form a right angle, so they are complements.

The measures of the two angles are

and


Find the measure of each angle.

1.1 Example 2(b) Finding Measures of Complementary and Supplementary Angles (page 3)

The two angles form a straight angle, so they are supplements.

The measures of the two angles are

and


Perform each calculation.

1.1 Example 3 Calculating with Degrees, Minutes, and Seconds (page 4)

(a)

(b)


1.1 Example 4 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds (page 5)

(a) Convert 105°20′32″ to decimal degrees.

(b) Convert 85.263° to degrees, minutes, and seconds.


Find the angles of least possible positive measure coterminal with each angle.

1.1 Example 5 Finding Measures of Coterminal Angles

(page 6)

(a) 1106°

(b) –150°

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 1106° is coterminal with an angle of 26°.

An angle of –150° is coterminal with an angle of 210°.


1.1 Example 5 Finding Measures of Coterminal Angles

(cont.)

(c) –603°

An angle of –603° is coterminal with an angle of 117°.


1.1 Example 6 Analyzing the Revolutions of a CD Player

(page 7)

A wheel makes 270 revolutions per minute. Through how many degrees will a point on the edge of the wheel move in 5 sec?

The wheel makes 270 revolutions in one minute or revolutions per second.

In five seconds, the wheel makesrevolutions.

Each revolution is 360°, so a point on the edge of the wheel will move


Angles1.2Geometric Properties ▪ Triangles


Find the measures of angles 1, 2, 3, and 4 in the figure, given that lines m and n are parallel.

1.2 Example 1 Finding Angle Measures (page 12)

Angles 2 and 3 are interior angles on the same side of the transversal, so they are supplements.


1.2 Example 1 Finding Angle Measures (cont.)

Angles 1 and 2 have equal measure because they are vertical angles, and angles 1 and 4 have equal measure because they are alternate exterior angles.


The measures of two of the angles of a triangle are 33° and 26°. Find the measure of the third angle.

1.2 Example 2 Finding Angle Measures (page 12)

The sum of the measures of the angles of a triangle is 360°.

Let x = the measure of the third angle.

The third angle measures 121°.


In the figure, triangles DEF and GHI are similar. Find the measures of angles G and I.

1.2 Example 3 Finding Angle Measures in Similar Triangles

(page 14)

The triangles are similar, so the corresponding angles have the same measure.


Given that triangle MNP and triangle QSR are similar, find the lengths of the unknown sides of triangle QSR.

1.2 Example 4 Finding Side Lengths in Similar Triangles

(page 15)

The triangles are similar, so the lengths of the corresponding sides are proportional.

PM corresponds to RQ.

PN corresponds to RS.

MN corresponds to QS.


1.2 Example 4 Finding Side Lengths in Similar Triangles

(cont.)


Samir wants to know the height of a tree in a park near his home. The tree casts a 38-ft shadow at the same time as Samir, who is 63 in. tall, casts a 42-in. shadow. Find the height of the tree.

1.2 Example 5 Finding the Height of a Flagpole (page 15)

Let x = the height of the tree

The tree is 57 feet tall.


Trigonometric Functions1.3Trigonometric Functions ▪ Quadrantal Angles


The terminal side of an angle θ in standard position passes through the point (12, 5). Find the values of the six trigonometric functions of angle θ.

1.3 Example 1 Finding Function Values of an Angle (page 23)

x = 12 and y = 5.

13


The terminal side of an angle θ in standard position passes through the point (8, –6). Find the values of the six trigonometric functions of angle θ.


x = 8 and y = –6.

10 6


1.3 Example 2 Finding Function Values of an Angle (cont.)


Find the values of the six trigonometric functions of angle θ in standard position, if the terminal side of θ is defined by 3x – 2y = 0, x ≤ 0.


Since x ≤ 0, the graph of the line 3x – 2y = 0 is shown to the left of the y-axis.

Find a point on the line:Let x = –2. Then

A point on the line is (–2, –3).


1.3 Example 3 Finding Function Values of an Angle (cont.)


Find the values of the six trigonometric functions of a 360° angle.

1.3 Example 4(a) Finding Function Values of Quadrantal Angles (page 26)

The terminal side passes through (2, 0). So x = 2 and y = 0 and r = 2.


Find the values of the six trigonometric functions of an angle θ in standard position with terminal side through (0, –5).

1.3 Example 4(b) Finding Function Values of Quadrantal Angles (page 26)

x = 0 and y = –5 and r = 5.


Using the Definitions of the Trigonometric Functions1.4Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities


Find each function value.

1.4 Example 1 Using the Reciprocal Identities (page 31)

(a) tan θ, given that cot θ = 4.

(b) sec θ, given that

tan θ is the reciprocal of cot θ.

sec θ is the reciprocal of cos θ.


Determine the signs of the trigonometric functions of an angle in standard position with the given measure.


(a) 54° (b) 260° (c) –60°

(a) A 54º angle in standard position lies in quadrant I, so all its trigonometric functions are positive.

(b) A 260º angle in standard position lies in quadrant III, so its sine, cosine, secant, and cosecant are negative, while its tangent and cotangent are positive.

(c) A –60º angle in standard position lies in quadrant IV, so cosine and secant are positive, while its sine, cosecant, tangent, and cotangent are negative.


Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions.

1.4 Example 3 Identifying the Quadrant of an Angle (page 33)

(a) tan θ > 0, csc θ < 0

(b) sin θ > 0, csc θ > 0

tan θ > 0 in quadrants I and III, while csc θ < 0 in quadrants III and IV. Both conditions are met only in quadrant III.

sin θ > 0 in quadrants I and II, as is csc θ. Both conditions are met in quadrants I and II.


Decide whether each statement is possible or impossible.

1.4 Example 4 Deciding Whether a Value is in the Range of a Trigonometric Function (page 33)

(a) cot θ = –.999 (b) cos θ = –1.7 (c) csc θ = 0

(a) cot θ = –.999 is possible because the range of cot θ is

(b) cos θ = –1.7 is impossible because the range of cos θ is [–1, 1].

(c) csc θ = 0 is impossible because the range of csc θ is


Angle θ lies in quadrant III, and Find the values of the other five trigonometric functions.

1.4 Example 5 Finding All Function Values Given One Value and the Quadrant (page 33)

Since and θ lies in quadrant III, then x = –5 and y = –8.


1.4 Example 5 Finding All Function Values Given One Value and the Quadrant (cont.)


1.4 Example 6 Finding Other Function Values Given One Value and the Quadrant (page 35)

Find cos θ and tan θ given that sin θ andcos θ > 0.

Reject the negative root.


1.4 Example 6 Finding Other Function Values Given One Value and the Quadrant (cont.)


1.4 Example 7 Finding Other Function Values Given One Value and the Quadrant (page 36)

Find cot θ and csc θ given that cos θ andθ is in quadrant II.

Since θ is in quadrant II, cot θ < 0 and csc θ > 0.

Reject the negative root:


1.4 Example 7 Finding Other Function Values Given One Value and the Quadrant (cont.)

1 Trigonometric Functions © 2008 Pearson Addison-Wesley. All rights reserved.

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measures of angles

angle page

angle measures page

supplementary angles

interior angles

angle relationships