7 Trigonometric Identities and Equations Sections 7 1

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7 Trigonometric Identities and Equations Sections 7. 1– 7. 4 © 2008 Pearson Addison-Wesley.

7 Trigonometric Identities and Equations Sections 7. 1– 7. 4 © 2008 Pearson Addison-Wesley. All rights reserved

7 Trigonometric Identities and Equations 7. 1 Fundamental Identities 7. 2 Verifying Trigonometric Identities

7 Trigonometric Identities and Equations 7. 1 Fundamental Identities 7. 2 Verifying Trigonometric Identities 7. 3 Sum and Difference Identities 7. 4 Double-Angle and Half-Angle Identities 2 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 1 Trigonometric Identities Fundamental Identities ▪ Using the Fundamental Identities 3 Copyright ©

7. 1 Trigonometric Identities Fundamental Identities ▪ Using the Fundamental Identities 3 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant

7. 1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (page 643) If value. and is in quadrant IV, find each function (a) In quadrant IV, is negative. 4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant

7. 1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (cont. ) If value. and is in quadrant IV, find each function (b) (c) 5 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 1 Example 2 Expressing One Function in Terms of Another (page 644) 6

7. 1 Example 2 Expressing One Function in Terms of Another (page 644) 6 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 1 Example 3 Rewriting an Expression in Terms of Sine and Cosine (page

7. 1 Example 3 Rewriting an Expression in Terms of Sine and Cosine (page 645) Write in terms of then simplify the expression. and , and 7 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2 Verifying Trigonometric Identities Verifying Identities by Working With One Side ▪ Verifying

7. 2 Verifying Trigonometric Identities Verifying Identities by Working With One Side ▪ Verifying Identities by Working With Both Sides 8 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2 Example 1 Verifying an Identity (Working With One Side) (page 650) Verify

7. 2 Example 1 Verifying an Identity (Working With One Side) (page 650) Verify that is an identity. Left side of given equation Right side of given equation 9 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2 Example 2 Verifying an Identity (Working With One Side) (page 650) Verify

7. 2 Example 2 Verifying an Identity (Working With One Side) (page 650) Verify that is an identity. Simplify. 10 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2 Example 3 Verifying an Identity (Working With One Side) (page 651) Verify

7. 2 Example 3 Verifying an Identity (Working With One Side) (page 651) Verify that is an identity. Simplify. Factor. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 11

7. 2 Example 4 Verifying an Identity (Working With One Side) (page 651) Verify

7. 2 Example 4 Verifying an Identity (Working With One Side) (page 651) Verify that is an identity. Multiply by 1 in the form 12 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2 Example 5 Verifying an Identity (Working With Both Sides) (page 652) Verify

7. 2 Example 5 Verifying an Identity (Working With Both Sides) (page 652) Verify that identity. is an Working with the left side: Multiply by 1 in the form Simplify. 13 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2 Example 5 Verifying an Identity (Working With Both Sides) (cont. ) Working

7. 2 Example 5 Verifying an Identity (Working With Both Sides) (cont. ) Working with the right side: Factor the numerator. Distributive property. Factor the denominator. Simplify. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 14

7. 2 Example 5 Verifying an Identity (Working With Both Sides) (cont. ) So,

7. 2 Example 5 Verifying an Identity (Working With Both Sides) (cont. ) So, the identity is verified. 15 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2 Example 6 Applying a Pythagorean Identity to Radios (page 653) FM radio

7. 2 Example 6 Applying a Pythagorean Identity to Radios (page 653) FM radio stations broadcast at higher frequencies than AM stations. For a certain classical music FM station, the energy stored in the inductor is given by and the energy in the capacitor is given by Write a simplified expression for E(t), the total energy in the circuit. 16 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2 Example 6 Applying a Pythagorean Identity to Radios (cont. ) Factor. 17

7. 2 Example 6 Applying a Pythagorean Identity to Radios (cont. ) Factor. 17 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Sum and Difference Identities Cosine Sum and Difference Identities ▪ Cofunction Identities

7. 3 Sum and Difference Identities Cosine Sum and Difference Identities ▪ Cofunction Identities ▪ Sine and Tangent Sum and Difference Identities 18 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 1 Finding Exact Cosine Function Values (page 659) Find the exact

7. 3 Example 1 Finding Exact Cosine Function Values (page 659) Find the exact value of each expression. (a) 19 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 1 Finding Exact Cosine Function Values (cont. ) (b) (c) 20

7. 3 Example 1 Finding Exact Cosine Function Values (cont. ) (b) (c) 20 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 2 Using Cofunction Identities to Find θ (page 660) Find an

7. 3 Example 2 Using Cofunction Identities to Find θ (page 660) Find an angle θ that satisfies each of the following. 21 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 3 Reducing cos (A – B) to a Function of a

7. 3 Example 3 Reducing cos (A – B) to a Function of a Single Variable (page 661) Write cos(90° + θ) as a trigonometric function of θ alone. 22 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 4 Finding Exact Sine and Tangent Function Values (page 663) Find

7. 3 Example 4 Finding Exact Sine and Tangent Function Values (page 663) Find the exact value of each expression. (a) 23 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 4 Finding Exact Sine and Tangent Function Values (cont. ) (b)

7. 3 Example 4 Finding Exact Sine and Tangent Function Values (cont. ) (b) 24 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 4 Finding Exact Sine and Tangent Function Values (cont. ) (c)

7. 3 Example 4 Finding Exact Sine and Tangent Function Values (cont. ) (c) 25 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 5 Writing Functions as Expressions Involving Functions of θ (page 664)

7. 3 Example 5 Writing Functions as Expressions Involving Functions of θ (page 664) Write each function as an expression involving functions of θ. (a) (b) (c) 26 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 6 Finding Function Values and the Quadrant of A – B

7. 3 Example 6 Finding Function Values and the Quadrant of A – B (page 665) Suppose that A and B are angles in standard position with and Find each of the following. (c) the quadrant of A – B. 27 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 6 Finding Function Values and the Quadrant of A – B

7. 3 Example 6 Finding Function Values and the Quadrant of A – B (cont. ) The identity for sin(A – B) requires sin A, cos A, sin B, and cos B. The identity for tan(A – B) requires tan A and tan B. We must find sin A, tan A, cos B and tan B. Because A is in quadrant III, sin A is negative and tan A is positive. Because B is in quadrant IV, cos B is positive and tan B is negative. 28 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 6(a) Finding Function Values and the Quadrant of A – B

7. 3 Example 6(a) Finding Function Values and the Quadrant of A – B (cont. ) To find sin A and cos B, use the identity 29 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 6(b) Finding Function Values and the Quadrant of A – B

7. 3 Example 6(b) Finding Function Values and the Quadrant of A – B (cont. ) To find tan A and tan B, use the identity 30 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 6(c) Finding Function Values and the Quadrant of A – B

7. 3 Example 6(c) Finding Function Values and the Quadrant of A – B (cont. ) From parts (a) and (b), sin (A − B) < 0 and tan (A − B) < 0. The only quadrant in which the values of both the sine and the tangent are negative is quadrant IV, so (A − B) is in quadrant IV. 31 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 7 Applying the Cosine Difference Identity to Voltage (page 666) Because

7. 3 Example 7 Applying the Cosine Difference Identity to Voltage (page 666) Because household current is supplied at different voltages in different countries, international travelers often carry electrical adapters to connect items they have brought from home to a power source. The voltage V in a typical European 220 -volt outlet can be expressed by the function (a) European generators rotate at precisely 50 cycles per second. Determine ω for these electric generators. Each cycle is radians at 50 cycles per second. 32 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 7 Applying the Cosine Difference Identity to Voltage (cont. ) (b)

7. 3 Example 7 Applying the Cosine Difference Identity to Voltage (cont. ) (b) What is the maximum voltage in the outlet? The maximum value of is 1. The maximum voltage in the outlet is 33 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 3 Example 7 Applying the Cosine Difference Identity to Voltage (cont. ) (c)

7. 3 Example 7 Applying the Cosine Difference Identity to Voltage (cont. ) (c) Determine the least positive value of that the graph of in radians so is the same as Using the sum identity for cosine gives 34 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Double-Angle and Half-Angle Identities Double-Angle Identities ▪ Product-to-Sum and Sum-to -Product Identities

7. 4 Double-Angle and Half-Angle Identities Double-Angle Identities ▪ Product-to-Sum and Sum-to -Product Identities ▪ Half-Angle Identities 35 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 1 Finding Function Values of 2θ Given Information About θ (page

7. 4 Example 1 Finding Function Values of 2θ Given Information About θ (page 673) The identity for sin 2θ requires cos θ. 36 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 1 Finding Function Values of 2θ Given Information About θ (cont.

7. 4 Example 1 Finding Function Values of 2θ Given Information About θ (cont. ) 37 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 1 Finding Function Values of 2θ Given Information About θ (cont.

7. 4 Example 1 Finding Function Values of 2θ Given Information About θ (cont. ) Alternatively, find tan θ and then use the tangent double-angle identity. 38 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Example 2 Finding Function Values of θ Given Information About 2θ (page 673) 7.

Example 2 Finding Function Values of θ Given Information About 2θ (page 673) 7. 4 Find the values of the six trigonometric functions of θ if Use the identity to find sin θ: θ is in quadrant III, so sin θ is negative. 39 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 2 Finding Function Values of θ Given Information About 2θ (cont.

7. 4 Example 2 Finding Function Values of θ Given Information About 2θ (cont. ) Use the identity to find cos θ: θ is in quadrant III, so cos θ is negative. 40 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 2 Finding Function Values of θ Given Information About 2θ (cont.

7. 4 Example 2 Finding Function Values of θ Given Information About 2θ (cont. ) 41 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 3 Verifying a Double-Angle Identity (page 674) Verify that is an

7. 4 Example 3 Verifying a Double-Angle Identity (page 674) Verify that is an identity. Difference of squares So, the identity is verified. 42 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 4 Simplifying Expressions Using Double-Angle Identities (page 674) Simplify each expression.

7. 4 Example 4 Simplifying Expressions Using Double-Angle Identities (page 674) Simplify each expression. 43 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 5 Deriving a Multiple-Angle Identity (page 675) Write cos 3 x

7. 4 Example 5 Deriving a Multiple-Angle Identity (page 675) Write cos 3 x in terms of cos x. Distributive property. Simplify. 44 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 6 Determining Wattage Consumption (page 676) If a toaster is plugged

7. 4 Example 6 Determining Wattage Consumption (page 676) If a toaster is plugged into a common household outlet, the wattage consumed is not constant. Instead it varies at a high frequency according to the model The voltage V in a typical European outlet can be expressed by the function. Find the maximum wattage of a European light bulb with R = 686. 45 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 6 Determining Wattage Consumption (cont. ) Substituting the given values in

7. 4 Example 6 Determining Wattage Consumption (cont. ) Substituting the given values in the wattage equation gives The maximum value of maximum wattage is is 1, so the 46 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 7 Using a Product-to-Sum Identity (page 677) Write 6 sin 40°

7. 4 Example 7 Using a Product-to-Sum Identity (page 677) Write 6 sin 40° sin 15° as the sum or difference of two functions. 47 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 8 Using a Sum-to-Product Identity (page 677) Write cos 3θ +cos

7. 4 Example 8 Using a Sum-to-Product Identity (page 677) Write cos 3θ +cos 7θ as a product of two functions. 48 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 9 Using a Half-Angle Identity to Find an Exact Value (page

7. 4 Example 9 Using a Half-Angle Identity to Find an Exact Value (page 679) Find the exact value of sin 22. 5° using the half-angle identity for sine. 49 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 10 Using a Half-Angle Identity to Find an Exact Value (page

7. 4 Example 10 Using a Half-Angle Identity to Find an Exact Value (page 679) Find the exact value of tan 75° using the identity 50 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 11 Finding Function Values of s 2 Given Information About s

7. 4 Example 11 Finding Function Values of s 2 Given Information About s (page 680) The angle associated with is positive while lies in quadrant II since are negative. 51 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 11 Finding Function Values of s 2 Given Information About s

7. 4 Example 11 Finding Function Values of s 2 Given Information About s (page 680) 52 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 4 Example 12 Simplifying Expressions Using the Half-Angle Identities (page 680) Simplify each

7. 4 Example 12 Simplifying Expressions Using the Half-Angle Identities (page 680) Simplify each expression. This matches part of the identity for . Substitute 8 x for A: 53 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.