Download presentation
Presentation is loading. Please wait.
Published byBarry Rice Modified over 8 years ago
1
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 5 Trigonometric Identities
2
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2 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 5.5Double-Angle Identities 5.6Half-Angle Identities 5 Trigonometric Identities
3
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Double-Angle Identities 5.5 Double-Angle Identities ▪ An Application ▪ Product-to-Sum and Sum-to-Product Identities
4
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Double-Angle Identities We can use the cosine sum identity to derive double-angle identities for cosine. Cosine sum identity
5
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Double-Angle Identities There are two alternate forms of this identity.
6
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Double-Angle Identities We can use the sine sum identity to derive a double-angle identity for sine. Sine sum identity
7
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 Double-Angle Identities We can use the tangent sum identity to derive a double-angle identity for tangent. Tangent sum identity
8
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Double-Angle Identities
9
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 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. Now find cos2θ, using the first double-angle identity for cosine (any of the three forms may be used).
10
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Example 1 FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ (cont.) Now find tan θ and then use the tangent double- angle identity.
11
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 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.
12
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 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.
13
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 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. Example 2
14
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 Example 3 VERIFYING A DOUBLE-ANGLE IDENTITY Quotient identity Verify that is an identity. Double-angle identity
15
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Example 4 SIMPLIFYING EXPRESSIONS USING DOUBLE-ANGLE IDENTITIES Simplify each expression. Multiply by 1. cos2A = cos 2 A – sin 2 A
16
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 Example 5 DERIVING A MULTIPLE-ANGLE IDENTITY Write sin 3x in terms of sin x. Sine sum identity Double-angle identities
17
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 Product-to-Sum Identities
18
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 Sum-to-Product Identities
19
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 Half-Angle Identities
Similar presentations
© 2024 SlidePlayer.com Inc.
All rights reserved.