Download 7.3 Double-Angle, Half-Angle, and Product

7.3 Double-Angle, Half-Angle,
and Product-Sum Formulas
Double Angle Formulas
sin 2 x  2sin x cos x
cos 2 x  cos x  sin x
2
2
 2 cos x  1
2
 1  2sin x
2 tan x
tan 2 x 
2
1  tan x
2
Simplify the expression by using a
double angle formula.
a) 2sin(12⁰)cos(12⁰)
b) cos217⁰ - sin217⁰
• If cosx=-2/3 and x is in Quadrant II, find sin2x
and cos2x.
• If csc x = -4, and x is in quadrant III, find sin2x
and cos2x.
Proving Identities
2cot x
 sin 2 x
2
1  cot x
Half Angle Formulas
x
1  cos x
sin  
2
2
x
1  cos x
cos  
2
2
x
1  cos x
tan  
2
1  cos x
1  cos x

sin x
sin x

1  cos x
Using Half-Angle Formulas
0
sin 22.5
• Find tan(u/2) if sinu=2/5 and u is in Quadrant
II.
Product-to-Sum Formulas
1
sin x cos y  sin( x  y)  sin( x  y) 
2
1
cos x sin y  sin( x  y)  sin( x  y) 
2
cos x cos y 
1
cos( x  y)  cos( x  y)
2
1
sin x sin y  cos( x  y)  cos( x  y) 
2
• Express sin3xsin5x as a sum of trigonometric
functions.
Sum-to-Product Formulas
x y
x y
sin x  sin y  2sin
cos
2
2
x y
x y
sin x  sin y  2 cos
sin
2
2
x y
x y
cos x  cos y  2 cos
cos
2
2
x y
x y
cos x  cos y  2sin
sin
2
2
• Write sin7x+sin3x as a product.
Proving an Identity
sin 3x  sin x
 tan x
cos 3x  cos x