5.5
Double–Angle and HalfAngle Formulas
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Double Angle/Multiple – Angle
Formulas
Double Angle Derivations:
sin 2u = sin (u + u) = sin u cos u + cos u sin u 2sinu cosu
cos 2u = cos (u + u) = cos u cos u– sin u sin u  cos2u– sin2u
tan 2u =tan (u+u) = (tanu +tanu)/(1-tanutanu) 2tanu/(1 – tan22u)
Example 1 – Solving a Double –Angle Equation
by first rewriting any double angles into “single
angles”
Solve for x in [0,2 ):
a) 2 cos x + sin 2x = 0
2 cos x + 2 sinx cosx = 0
2 cos x ( 1 + sinx) = 0
2 cos x = 0 1 + sinx = 0
cos x = 0
sin x = -1
x= /2, 3 /2
x = 3 /2
b) sin 2x sin x = cos x
2 sinx cos x sin x – cos x = 0
2 sin2x cos x – cos x = 0
cos x (2sin2x-1) = 0
cos x = 0 2sin2x-1 = 0
sin x = 1/ 2
x = /2, 3 /2
x = /4, 3 /4
5 /4, 7 /4
c) Solve for x in (- , ) tan 2x – cot x = 0
You try this one!!
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Example 2 – Solving a Multiple–Angle Equation
by first rewriting any multiple angles into “single
angles”
Solve in (- , ): sin 4x = -2 sin 2x
Rewrite this one as:
sin 2(2x) + 2 sin 2x = 0
2sin 2x cos 2x + 2 sin 2x = 0
2 sin 2x [cos 2x + 1]
=0
2 sin 2x = 0
cos 2x + 1 = 0
sin 2x = 0
cos 2x = -1
2x = 0 + 2n , + 2n
2x = + 2n
x = 0 + n , /2 + n
x = /2 + n
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Half–Angle Formulas
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Example 3 – Solving a Trigonometric Equation by
rewriting all half-angles into “single angles”
Find all solutions in the interval [0, 2 ).
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Example 4 – Using Half-Angle Formulas to find
exact values of uncommon angles
Use the half-angle formulas to determine the exact values of the
sine, cosine, and tangent of the following angles:
a) 75˚
b) π/12
c) 3π/8
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