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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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reciprocal identities
Reciprocal Identities

Sin u =

Cos u =

Tan u =

Csc u =

Sec u =

Cot u =

quotient identities
Quotient Identities

Tan u =

Cot u =

pythagorean identities
Pythagorean Identities

sin2 u + cos2 u = 1

1 + tan2 u = sec2 u

1 + cot2 u = csc2 u

cofunction identities
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

odd even identities
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

what are they good for
What are they good for?

One use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions.

example 1 using identities to evaluate a function
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.

example 2 simplifying a trigonometric expression
Example 2: Simplifying a Trigonometric Expression

Simplify sin x cos2 x – sin x

example 3 verifying a trigonometric identity
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:
example 4 factoring trigonometric expressions
Example 4: Factoring Trigonometric Expressions

Factor

  • sec2 x – 1
  • 4tan2 x + tan x – 3
  • csc2 x – cot x – 3