Using Fundamental Identities
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Transcript of Using Fundamental Identities
Using Fundamental Identities5.1
Reciprocal Identities
Sin u =
Cos u =
Tan u =
Csc u =
Sec u =
Cot u =
ucsc
1
usec
1
ucot
1
usin
1
ucos
1
utan
1
Quotient Identities
Tan u =
Cot u =
u
u
cos
sin
u
u
sin
cos
Pythagorean Identities
sin2 u + cos2 u = 1
1 + tan2 u = sec2 u
1 + cot2 u = csc2 u
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
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?
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
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
Simplify sin x cos2 x – sin x
Example 3: Verifying a Trigonometric Identity
A) Determine whether the equation appears to be an identity. Cos 3x = 4 cos3 x – 3cos x.
B) Verify the identity: xx
x
x
xcsc
sin
cos
cos1
sin
Example 4: Factoring Trigonometric ExpressionsFactor
A)sec2 x – 1
B)4tan2 x + tan x – 3
C)csc2 x – cot x – 3
Example 5: Simplifying a Trigonometric Expression
Simplify: sin t + cot t∙cos t