Lecture 16 section 6.2 sum-difference identities

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Transcript of Lecture 16 section 6.2 sum-difference identities

  • 1. MATH 107 Section 6.2 Sum and Difference Formulas

2. 2 2011 Pearson Education, Inc. All rights reserved SUM AND DIFFERENCE FORMULAS FOR COSINE ( ) ( ) cos cos cos sin sin cos cos cos sin sin u v u v u v u v u v u v + = = + 3. 3 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Difference Identity for Cosine Find the exact value of by using 12 = 3 4 . Solution cos cos 12 3 4 = = 1 2 2 2 + 3 2 2 2 = 2 4 + 6 4 = 2 + 6 4 = cos 3 cos 4 + sin 3 sin 4 12 cos 4. Example 4 2011 Pearson Education, Inc. All rights reserved Find the exact value of cos15 . 5. 5 2011 Pearson Education, Inc. All rights reserved SUM AND DIFFERENCE IDENTITIES FOR SINE ( ) ( ) vuvuvu vuvuvu sincoscossinsin sincoscossinsin = +=+ 6. 19 Find the exact value of sin . 12 (Answer on next slide.) 7. 19 Find the exact value of sin . 12 19 16 3 4 sin sin sin 12 12 12 3 4 = + = + 8. 8 2011 Pearson Education, Inc. All rights reserved EXAMPLE 5 Using the Sum Formula for Sine Find the exact value of sin63 cos27 + cos63sin27 without using a calculator. ( )sin63cos27 cos63sin27 sin 63 27+ = + ( )sin 90= = 1 Solution This expression is the right side of the sum identity for sin (u + v), where u = 63 and v = 27. 9. Example 9 2011 Pearson Education, Inc. All rights reserved Find the exact value of sin75 10. 10 2011 Pearson Education, Inc. All rights reserved .sin 2 cos vv = BASIC COFUNCTION IDENTITIES If v is any real number or angle measured in radians, then If angle v is measured in degrees, then replace by 90 in these identities. 2 .cos 2 sin vv = 11. 11 2011 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using Cofunction Identities Prove that for any real number x, tan cot . 2 x x = Solution tan 2 x = sin 2 x cos 2 x cos sin x x = cot x= 12. 12 2011 Pearson Education, Inc. All rights reserved SUM AND DIFFERENCE IDENTITIES FOR TANGENT ( ) ( ) tan tan tan 1 tan tan tan tan tan 1 tan tan u v u v u v u v u v u v = + + + = 13. 13 2011 Pearson Education, Inc. All rights reserved EXAMPLE 11 Verifying an Identity Verify the identity .tan x( ) = tan x Solution Apply the difference identity. ( ) tan tan tan 1 tan tan x x x = + ( ) 0 tan tan 1 0 tan x x x = + ( )tan tanx x = Therefore, the given equation is an identity.