Chapter 8 The Trigonometric Functions Copyright 2014 2010

Chapter Outline • Radian Measure of Angles • The Sine and the Cosine •

Section 8. 4 The Tangent and Other Trigonometric Functions Copyright © 2014, 2010, 2007

Section Outline • Other Trigonometric Functions • Other Trigonometric Identities • Applications of Tangent

Other Trigonometric Functions Certain functions involving the sine and cosine functions occur so frequently

Other Trigonometric Identities (3) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 6

Applications of Tangent EXAMPLE Find the width of a river at points A and

Applications of Tangent CONTINUED We convert 40° into radians. We find that 40° =

Derivative Rules for Tangent The derivative of tan t can be expressed in two

Differentiating Tangent EXAMPLE Differentiate. SOLUTION From equation (5) we find that Copyright © 2014,

The Graph of Tangent tan t is defined for all t except where cos

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Chapter Outline • Radian Measure of Angles • The Sine and the Cosine • Differentiation and Integration of sin t and cos t • The Tangent and Other Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Section Outline • Other Trigonometric Functions • Other Trigonometric Identities • Applications of Tangent • Derivative Rules for Tangent • Differentiating Tangent • The Graph of Tangent Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Other Trigonometric Functions Certain functions involving the sine and cosine functions occur so frequently in applications that they have been given special names. The tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) are such functions and are defined as follows: Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 5

Applications of Tangent EXAMPLE Find the width of a river at points A and B if the angle BAC is 90°, the angle ACB is 40°, and the distance from A to C is 75 feet. r SOLUTION Let r denote the width of the river. Then equation (3) implies that Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 7

Applications of Tangent CONTINUED We convert 40° into radians. We find that 40° = (π/180)40 radians ≈ 0. 7 radians, and tan(0. 7) ≈ 0. 84229. Hence Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 8

Derivative Rules for Tangent The derivative of tan t can be expressed in two equivalent ways: (4) Combining (4) with the chain rule, we have (5) Integration: Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 9

The Graph of Tangent tan t is defined for all t except where cos t = 0. (We cannot have zero in the denominator of sin t/ cos t. ) The graph of tan t is sketched in Fig. 5. Note that tan t is periodic with period π. Fig. 5 Graph of Tangent Function Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 11