5 Trigonometric Identities Copyright 2013 2009 2005 Pearson

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5 Trigonometric Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

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

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 ▪ An Application ▪ Product-to. Sum and Sum-to-Product Identities Copyright

5. 5 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

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,

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 doubleangle identity

Double-Angle Identities We can use the sine sum identity to derive a doubleangle 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

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

Double-Angle Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8

Example 1 Given 2θ. FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ and

Example 1 Given 2θ. FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ and sin θ < 0, find sin 2θ, cos 2θ, and tan Now use the double-angle identity for sine. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9

Example 2 FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ Find the values

Example 2 FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ Find the values of the six trigonometric functions of θ if Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10

Example 3 Verify that VERIFYING A DOUBLE-ANGLE IDENTITY is an identity. Quotient identity Double-angle

Example 3 Verify that VERIFYING A DOUBLE-ANGLE IDENTITY is an identity. Quotient identity Double-angle identity Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11

Example 4 SIMPLIFYING EXPRESSIONS USING DOUBLE-ANGLE IDENTITIES Simplify each expression. cos 2 A =

Example 4 SIMPLIFYING EXPRESSIONS USING DOUBLE-ANGLE IDENTITIES Simplify each expression. cos 2 A = cos 2 A – sin 2 A Multiply by 1. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12

Example 5 DERIVING A MULTIPLE-ANGLE IDENTITY Write sin 3 x in terms of sin

Example 5 DERIVING A MULTIPLE-ANGLE IDENTITY Write sin 3 x in terms of sin x. Sine sum identity Double-angle identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13

Assignment Page 231 #3 -51 M 3 Copyright © 2013, 2009, 2005 Pearson Education,

Assignment Page 231 #3 -51 M 3 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14