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Chapter 5 Trigonometric Identities

Chapter 5 Trigonometric Identities. Trigonometric Identities. Quotient Identities. Reciprocal Identities. Pythagorean Identities. sin 2 q + cos 2 q = 1. tan 2 q + 1 = sec 2 q. cot 2 q + 1 = csc 2 q. sin 2 q = 1 - cos 2 q. tan 2 q = sec 2 q - 1. cot 2 q = csc 2 q - 1.

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Chapter 5 Trigonometric Identities

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  1. Chapter 5 Trigonometric Identities

  2. Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin2q + cos2q = 1 tan2q + 1 = sec2q cot2q + 1 = csc2q sin2q = 1 - cos2q tan2q = sec2q - 1 cot2q = csc2q - 1 cos2q = 1 - sin2q 5.4.3

  3. Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. b) a) 5.4.5

  4. Simplifing Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x d) 5.4.6

  5. Simplify a Trigonometric Expression

  6. Simplify a Trigonometric Expression

  7. Simplify a Trigonometric Expression

  8. Simplify a Trigonometric Expression

  9. Verifying an Identity When verifying a trigonometric identity, we work with one side of the equation until we make it look exactly like the other side of the equation. When working with the one side of the equation, we can multiply that side by 1 (or any fraction that is equal to 1), rearrange terms within the side we are working on, combine or take apart fractions that are on the side we are working on, and replace terms with equal terms. The one thing that we cannot do is move things from one side of the equation to the other.

  10. Verifying an Identity Steps in Verifying Identities 1. Start with the more complex side of the identity and work with it exclusively to transform the expression into the simpler side of the identity. 2. Look for algebraic simplifications: • Do any multiplying , factoring, or squaring which is • obvious in the expression. • Reduce two terms to one, either add two terms or • factor so that you may reduce. 3. Look for trigonometric simplifications: • Look for familiar trig relationships. • If the expression contains squared terms, think • of the Pythagorean Identities. • Transform each term to sine or cosine, if the • expression cannot be simplified easily using other ratios. 4. Keep the simpler side of the identity in mind. 5.4.7

  11. Verifying an Identity Verify the following: a) sec x(1 + cos x) = 1 + sec x = sec x + sec x cos x = sec x + 1 1 + sec x L.S. = R.S. b) sec x = tan x csc x c) tan x sin x + cos x = sec x L.S. = R.S. L.S. = R.S. 5.4.8

  12. Verifying an Identity d) sin4x - cos4x = 1 - 2cos2x 1 - 2cos2x = (sin2x - cos2x)(sin2x + cos2x) = (1 - cos2x - cos2x) = 1 - 2cos2x L.S. = R.S. e) L.S. = R.S. 5.4.9

  13. Verifying an Identity f) L.S. = R.S. 5.4.10

  14. On Your OwnVerify the trigonometric identity. For this example, since the left side is more complicated, you should work with that side trying to simplify it.

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