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5 Trigonometric Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 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 © 2013, 2009, 2005 Pearson Education, Inc. 2 5.5 Double-Angle Identities Double-Angle Identities ▪ An Application ▪ Product-to-Sum and Sum-to-Product Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Double-Angle Identities We can use the cosine sum identity to derive double-angle identities for cosine. Cosine sum identity Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Double-Angle Identities There are two alternate forms of this identity. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Double-Angle Identities We can use the sine sum identity to derive a double-angle identity for sine. Sine sum identity Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Double-Angle Identities We can use the tangent sum identity to derive a double-angle identity for tangent. Tangent sum identity Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 Double-Angle Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Example 1 FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ Given and sin θ < 0, find sin 2θ, cos 2θ, and tan 2θ. Find Sin θ using the appropriate quadrant. Now use the double-angle identity for sine. 24 4 3 sin2 2sin cos 2 5 5 25 Now find cos2θ, using the first double-angle identity for cosine (any of the three forms may be used). 9 16 7 2 2 cos 2 cos sin 25 25 25 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 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 © 2013, 2009, 2005 Pearson Education, Inc. 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. 24 sin2 25 24 tan2 cos 2 7 7 25 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Example 2 FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ Find the values of the six trigonometric functions of θ if We must obtain a trigonometric function value of θ alone. θ is in quadrant II, so sin θ is positive. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 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 © 2013, 2009, 2005 Pearson Education, Inc. 13 Example 3 Verify that VERIFYING A DOUBLE-ANGLE IDENTITY is an identity. Quotient identity Double-angle identity Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 Example 4 SIMPLIFYING EXPRESSIONS USING DOUBLE-ANGLE IDENTITIES Simplify each expression. cos2A = cos2A – sin2A Multiply by 1. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Example 5 DERIVING A MULTIPLE-ANGLE IDENTITY Write sin 3x in terms of sin x. Sine sum identity Double-angle identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 Product-to-Sum Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 Sum-to-Product Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 Half-Angle Identities A 1 cos A cos 2 2 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19