Trig identities
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This document discusses trigonometric identities. It includes: - Pythagorean identities relating sine, cosine, and their reciprocals - Tangent/cotangent identities relating tangent and cotangent to sine and cosine - Fundamental trig identities for sine, tangent, cotangent, and secant - Examples of simplifying trig expressions using identities and of proving trig identities by manipulating expressions until they are equal.
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Trig identities
- 1. Trig Identities • Simplifying Trig Expressions • Proving Trig Identities
- 2. Pythagorean Identities: Reciprocal Identities: 1 csc sin 1 sec cos 1 cot tan x x x x x x = = = Tangent/Cotangent Identities: x x x x x x sin cos cot cos sin tan = = Fundamental Trig Identities 2 2 sin cos 1x x+ = 2 2 tan 1 secx x+ = 2 2 1 cot cscx x+ =
- 3. Example 1 Simplify the trig expression: cos tan sint t t+ Solution: cos tan sint t t+ = sin cos sin cos t t t t + ÷ 2 2 cos sin cos t t t + = 1 cost = sect= 2 sin cos cos t t t = +
- 4. Example 2 Simplify the expression: sin cot cosu u u+ Answer: cscu
- 5. Example 3 Simplify the expression: cos cos 1 sin 1 sin x x x x + = − + cos cos 1 sin 1 sin x x x x + − + cos (1 sin ) cos (1 sin ) (1 sin )(1 sin ) (1 sin )(1 sin ) x x x x x x x x + − + − + + − 2 2 cos sin cos cos sin cos 1 sin 1 sin x x x x x x x x + − = + − − 2 2cos 1 sin x x = − 2 2cos cos x x = 2 2sec cos x x = = Solution:
- 6. Tips for Proving Trig Identities Start with one side of the equation and manipulate it until it equals the other side. (Try the more complicated side first!) Look for chances to use identities and/or algebraic techniques (adding fractions, factoring, multiplying by a form of “1”, etc.) If you get stuck, try re-writing everything in terms of the sine and cosine. * Can also try working with each side of the equation separately until you obtain the same expression.
- 7. Example 4 Prove the identity: 1 1 2tan sec 1 sin 1 sin x x x x = − − +
- 8. Example 5 Verify the identity: 2 1 cos tan cos sec 1 θ θ θ θ + = − 1 cos LHS = cos θ θ + 1 cos cos cos θ θ θ = + sec 1θ= + 2 tan RHS = sec 1 θ θ − 2 sec 1 sec 1 θ θ − = − (sec 1)(sec 1) sec 1 θ θ θ + − = − sec 1θ= +