 # 7 Trigonometric Identities and Equations Sections 7 1

• Slides: 53  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 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 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 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 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 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. 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 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 (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 (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(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 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) 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) 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.     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 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 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 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 679) Find the exact value of tan 75° using the identity 50 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.   