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Review – Trigonometry
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Trigonometric Identities
Trigonometric Equations
Double Angle Identities
2
2
cos θ + sin θ = 1
2
2
1 + tan θ = sec θ
2
2
cot + 1 = csc θ
sin2θ = 2sinθcosθ
2
2
cos 2θ = cos θ - sin θ
2
= 1 – 2sin θ
2
= 2cos θ - 1
TRIGONOMETRIC IDENTIES
Prove the following trigonometric identities.
1.
cos x + cot x sin x
= 2 sin x
cot x
2.
2 cot x
= 2 cos 2 x
cot x + tan x
3.
1 + sin x 1 + sin x
=
1 − sin x
cos x
4.
1
cos x
−
= 2 csc 2 x − 1
1 − cos x 1 + cos x
5. sec x − tan x =
1 − sin x
cos x
TRIGONOMETRIC EQUATIONS
Solve the following equations if 0 < x < 2π.
1. 4cos2x – 3 = 0
2. 2sin3x = sinx
3. 2sin2x = 1 – cosx
4. 8sin2xcosx – 2cosx – 4sin2x + 1 = 0 Hint: use factoring by groupings
DOUBLE ANGLE IDENTITIES
1. If sin x = 2/3, x in QII, find sin 2x and cos 2x.
2. If cos x = −
5
, x in QIII, find sin 2x and cos 2x.
2
CIRCULAR FUNCTIONS
x2 + y2 = r2
cos θ =
x
r
sin θ =
x
r
r
y
θ
x
1. If sin θ =
1
, θ in QII, find the remaining trigonometric functions.
3
2. If tan θ =
5
, θ in QIII, find the remaining trigonometric functions.
7
3. If sec =
4
, θ in QI, find the remaining trigonometric functions.
3
tan θ =
y
x