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Section 7.4 trigonometric identities
This document contains derivations of trigonometric identities. It simplifies expressions involving trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant by rewriting them in terms of other trigonometric functions or common denominators. It also uses factoring and trigonometric angle addition and subtraction formulas to derive new identities from existing ones. The document derives over a dozen new trigonometric identities through algebraic manipulation and rearrangement of trigonometric expressions.
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Section 7.4 trigonometric identities
- 1. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. Section 7.4 Trigonometric Identities
- 2. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall.
- 3. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall.
- 4. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall.
- 5. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall.
- 6. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. tan (a) Simplify by rewriting each trigonometric function in terms of sine and cosine functions. sec sin 1 cos (b) Show that by multiplying the numerator and denominator by 1 cos 1+cos sin θ θ θ θ θ θ θ − = − sin tan sin coscos(a) sin 1sec cos 1 cos θ θ θ θθ θ θ θ = = × = ( ) ( ) ( ) sin 1 cossin (b) 1 cos 1 cos 1 cos θ θθ θ θ θ − = + + − ( ) 2 sin 1 cos 1 cos θ θ θ − = − ( ) 2 sin 1 cos sin θ θ θ − = 1 cos sin θ θ − =
- 7. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. (a) Simplify 1 1− sinu + 1 1+ sinu by rewriting the expression over a common denominator. (b) Simplify 1− cos2 v sinv + cosvsinv by factoring. ( ) ( ) ( ) ( ) ( ) ( ) 1 sin 1 sin 1 sin 1 1 (a) 1 sin 1 si sn 1 in u u uu u u + − + − + − + 2 1 sin 1 sin 1 sin u u u + + − = − = 2 cos2 u = 2sec2 u ( ) ( ) ( ) 2 1 cos 1 cos1 cos (b) sin cos sin sin 1 cos v vv v v v v v + −− = + + 1 cos 1 cos csc cot sin sin sin v v v v v v v − = = − = −
- 8. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall.
- 9. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. ( )sin cot tan secθ θ θ θ+ = ( ) cos sin sin cot tan sin sin cos θ θ θ θ θ θ θ θ + = + ÷ 2 2 2 sin cos sin cos cos cos θ θ θ θ θ θ + = + = 1 sec cos θ θ = =
- 10. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. sin csc cot 1 cos θ θ θ θ − = + 1 cos sin θ θ − = 1 cos csc cot sin sin θ θ θ θ θ − = − ( ) ( ) ( ) 1 cos 1 cos sin 1 cos θ θ θ θ − + = + ( ) 2 1 cos sin 1 cos θ θ θ − = + ( ) 2 sin sin 1 cos θ θ θ = + sin 1 cos θ θ = +
- 11. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. 2 2 2 sin tan tan cos cot θ θ θ θ θ − = − 2 2 2 2 sin sin sin tan cos coscos cot cos sin θ θ θ θ θ θθ θ θ θ − − = − − 2 2 sin cos sin cos sin cos cos sin θ θ θ θ θ θ θ θ − = − 2 2 sin cos sin sin cos sin cos cos θ θ θ θ θ θ θ θ − = × − ( ) ( ) 2 2 sin sin cos 1 cos sin cos 1 θ θ θ θ θ θ − = − 2 2 2 sin tan cos θ θ θ = =
- 12. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. cos 1 sin 1 sin cos θ θ θ θ + = − ( ) ( ) ( ) cos 1 sincos 1 sin 1 sin 1 sin θ θθ θ θ θ + = − − + ( ) 2 cos 1 sin 1 sin θ θ θ + = − ( ) 2 cos 1 sin cos θ θ θ + = 1 sin cos θ θ + =
- 13. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. 2 csc sin cot sin θ θ θ θ − = 1 sin csc sin sinsin sin sin sin θ θ θ θθ θ θ θ − ÷− = ÷ ÷ ÷ 2 2 2 2 2 1 sin cos cot sin sin θ θ θ θ θ − = = =
- 14. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall. 3 2 1 csc sin cosθ θ θ− = 3 3 sin 1 csc sin 1 sin θ θ θ θ − = − 2 2 1 sin cosθ θ= − =
- 15. Copyright © 2012Pearson Education, Inc. Publishing as Prentice Hall.