Transcript 5.5
5 Trigonometric Identities Copyright © 2009 Pearson Addison-Wesley 5.5-1 5 Trigonometric Identities 5.1 Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference Identities for Cosine 5.4 Sum and Difference Identities for Sine and Tangent 5.5 Double-Angle Identities 5.6 Half-Angle Identities Copyright © 2009 Pearson Addison-Wesley 5.5-2 5.5 Double-Angle Identities Double-Angle Identities ▪ An Application ▪ Product-to-Sum and Sum-to-Product Identities Copyright © 2009 Pearson Addison-Wesley 1.1-3 5.5-3 Double-Angle Identities We can use the cosine sum identity to derive double-angle identities for cosine. Cosine sum identity Copyright © 2009 Pearson Addison-Wesley 5.5-4 Double-Angle Identities There are two alternate forms of this identity. Copyright © 2009 Pearson Addison-Wesley 5.5-5 Double-Angle Identities We can use the sine sum identity to derive a double-angle identity for sine. Sine sum identity Copyright © 2009 Pearson Addison-Wesley 5.5-6 Double-Angle Identities We can use the tangent sum identity to derive a double-angle identity for tangent. Tangent sum identity Copyright © 2009 Pearson Addison-Wesley 5.5-7 Double-Angle Identities Copyright © 2009 Pearson Addison-Wesley 1.1-8 5.5-8 Example 1 FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ and sin θ < 0, find sin 2θ, cos 2θ, and Given tan 2θ. The identity for sin 2θ requires sin θ. Any of the three forms may be used. Copyright © 2009 Pearson Addison-Wesley 1.1-9 5.5-9 Example 1 FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ (cont.) Now find tan θ and then use the tangent doubleangle identity. Copyright © 2009 Pearson Addison-Wesley 1.1-10 5.5-10 Example 1 FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ (cont.) Alternatively, find tan 2θ by finding the quotient of sin 2θ and cos 2θ. Copyright © 2009 Pearson Addison-Wesley 1.1-11 5.5-11 Example 2 FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ Find the values of the six trigonometric functions of θ if to find sin θ: Use the identity θ is in quadrant II, so sin θ is positive. Copyright © 2009 Pearson Addison-Wesley 1.1-12 5.5-12 Example 2 FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ (cont.) Use a right triangle in quadrant II to find the values of cos θ and tan θ. Use the Pythagorean theorem to find x. Copyright © 2009 Pearson Addison-Wesley 1.1-13 5.5-13 Example 3 VERIFYING A DOUBLE-ANGLE IDENTITY Verify that is an identity. Quotient identity Double-angle identity Copyright © 2009 Pearson Addison-Wesley 1.1-14 5.5-14 Example 4 SIMPLIFYING EXPRESSION DOUBLEANGLE IDENTITIES Simplify each expression. Multiply by 1. Copyright © 2009 Pearson Addison-Wesley 1.1-15 5.5-15 Example 5 DERIVING A MULTIPLE-ANGLE IDENTITY Write sin 3x in terms of sin x. Sine sum identity Double-angle identities Copyright © 2009 Pearson Addison-Wesley 1.1-16 5.5-16 Example 6 DETERMINING WATTAGE CONSUMPTION 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 where V is the voltage and R is a constant that measure the resistance of the toaster in ohms.* Graph the wattage W consumed by a typical toaster with R = 15 and in the window [0, .05] by [–500, 2000]. How many oscillations are there? *(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall, 1988.) Copyright © 2009 Pearson Addison-Wesley 1.1-17 5.5-17 Example 6 DETERMINING WATTAGE CONSUMPTION There are six oscillations. Copyright © 2009 Pearson Addison-Wesley 1.1-18 5.5-18 Product-to-Sum Identities The identities for cos(A + B) and cos(A – B) can be added to derive a product-to-sum identity for cosines. Copyright © 2009 Pearson Addison-Wesley 5.5-19 Product-to-Sum Identities Similarly, subtracting cos(A + B) from cos(A – B) gives a product-to-sum identity for sines. Copyright © 2009 Pearson Addison-Wesley 5.5-20 Product-to-Sum Identities Using the identities for sin(A + B) and sine(A – B) gives the following product-to-sum identities. Copyright © 2009 Pearson Addison-Wesley 5.5-21 Product-to-Sum Identities Copyright © 2009 Pearson Addison-Wesley 1.1-22 5.5-22 Example 7 USING A PRODUCT-TO-SUM IDENTITY Write 4 cos 75° sin 25° as the sum or difference of two functions. Copyright © 2009 Pearson Addison-Wesley 1.1-23 5.5-23 Sum-to-Product Identities Copyright © 2009 Pearson Addison-Wesley 1.1-24 5.5-24 Example 8 Write Copyright © 2009 Pearson Addison-Wesley USING A SUM-TO-PRODUCT IDENTITY as a product of two functions. 1.1-25 5.5-25