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Using Fundamental Identities

Using Fundamental Identities. 5.1. Reciprocal Identities. Sin u = Cos u = Tan u = Csc u = Sec u = Cot u =. Quotient Identities. Tan u = Cot u =. Pythagorean Identities. sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 + cot 2 u = csc 2 u. Cofunction Identities.

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Using Fundamental Identities

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  1. Using Fundamental Identities 5.1

  2. Reciprocal Identities Sin u = Cos u = Tan u = Csc u = Sec u = Cot u =

  3. Quotient Identities Tan u = Cot u =

  4. Pythagorean Identities sin2 u + cos2 u = 1 1 + tan2 u = sec2 u 1 + cot2 u = csc2 u

  5. Cofunction Identities sin (π/2 – u) = cos u csc (π/2 – u) = sec u cos (π/2 – u) = sin u sec (π/2 – u) = csc u tan (π/2 – u) = cot u cot (π/2 – u) = tan u

  6. Odd & Even Identities sin (-u) = -sin u csc (-u) = -csc u cos (-u) = cos u sec (-u) = sec u tan (-u) = -tan u cot (-u) = -cot u

  7. What are they good for? One use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions.

  8. Example 1: Using Identities to Evaluate a Function Use the values of sec u = -3/2 and tan u > 0 to find the values of all six trigonometric functions.

  9. Example 2: Simplifying a Trigonometric Expression Simplify sin x cos2 x – sin x

  10. Example 3: Verifying a Trigonometric Identity • Determine whether the equation appears to be an identity. Cos 3x = 4 cos3 x – 3cos x. • Verify the identity:

  11. Example 4: Factoring Trigonometric Expressions Factor • sec2 x – 1 • 4tan2 x + tan x – 3 • csc2 x – cot x – 3

  12. Example 5: Simplifying a Trigonometric Expression Simplify: sin t + cot t∙cos t

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