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Trigonometric Identities
Reciprocals
Quotients
Pythagorean
When simplifying expressions, or factoring expressions, look for common factors.
If none exist, look for related functions from the identities that will assist in the
simplification.
Examples:
1.
Substitute:
2.
Factor difference of
perfect squares.
Substitute:
3.
Version used:
Substitute:
4.
Multiply and express as a monomial:
Version used:
1) Simplify the expression (1-cos2x)(cscx) to a single trigonometric function.
2) Simplify
1  cos2 x
sec 2 x  1
Ans: cos2x
3) Simplify: cos2x + cos2x tan2x
Ans: cos2x + cos2x = 1
sin  cos

cos

sin 
4) Simplify:
1
sin 
Ans:
5) secθ (cosθ) – cos2θ
1
cos
Ans: 1 – cos2θ = sin2θ
Ans: Sin x
More Examples
2. Simplify the expression (1 – cos2x)(cscx) to a single trig functions
3. Simplify: cos2x + (cos2x)(tan2x)
4. Write this expression as a monomial with a single trig function:
(secθ)(cosθ) – cos2θ
6. Write as a monomial with a single trig function
a) sinθcotθ
b) sinθsecθ
c) secθcotθsinθ
d) secθsinθcscθ
e) cscθ(1- cos2θ)
f) sinθ(cot2θ + 1)
g) secθcosθ – cos2θ
Using Identities in Equation Solving
IF there is more than one trig function in the equation, identities are needed to
reduce the equation to a single trig function
1) Solve: 2cos2θ + 3sinθ – 3 = 0
2(1 – sin2 θ) + 3sin θ – 3 = 0
2 – 2sin2 θ + 3sin θ – 3 = 0
-2sin2 θ + 3sin θ – 1 = 0
replace using sin2 θ + cos2 θ= 1
distribute
combine like terms
0 = 2sin2 θ - 3sin θ + 1
move to other side to make a >0
0 = (2sinθ - 1)(sin θ – 1)
factor
sin θ = ½
sin θ = 1
t-chart and solve for sin θ
1) RA = 30
Quadrantal
do the steps to solve for θ
2) positive
θ = 90°
3) QI
Θ = RA
QII
θ = 180 – RA
Θ = 30°
θ = 180 – 30
Θ = 150°
Θ = {30°, 90°, 150°}
2) 2cos2θ – sinθ = 1
3) sec2θ – tanθ – 1 = 0
4) cosθ = secθ
5) 2sinθ = cscθ