Transcript Section 7.3 Double-Angle, Half-Angle and Product-Sum Formulas

Section 7.3
Double-Angle, Half-Angle and
Product-Sum Formulas
Objectives:
•To understand and apply the doubleangle formula.
•To understand and apply the half-angle
formula.
•To understand and apply product-sum
formula.
Class Work
1. Evaluate csc780
2. Find the exact value of tan165
Double-Angle Formulas
Formula for sine: sin 2x  2sin x cos x
Formula for cosine: cos 2 x  cos2 x  sin 2 x
 1  2sin 2 x
 2cos2 x  1
2 tan x
Formula for tangent: tan 2 x 
1  tan 2 x
2
cos x  
3
Ex 1. If
and x is in quadrant II, find
cos 2x, sin 2x, and tan 2x.
3
sin x 
5
Ex 2. If
and x is in quadrant I, then find
sin 2x, cos 2x, and tan 2x.
Class Work
6
cos x   and x is in quadrant II, find
11
3. If
sin 2x, cos 2x, and tan 2x.
Half-Angle Formulas
u
1  cos u
sin  
2
2
u
1  cos u
cos  
2
2
u 1  cos u
tan 
2
sin u
sin u

1  cos u
Ex 3. Find the exact value of sin 22.5.
Ex 4. Find tan

8
Ex 5. Find
u
tan
2
2
if sin u 
5
and u is in quad II.
Class Work
4.
u
u
u
Find sin , cos , and tan
2
2
2
quad IV.
if
cos u 
2
3
and u is in
Simplify the expression by using a double-angle
or half-angle formula.
5. 2sin35 cos35

6.
cos
7.
1  cos 4
2
2
2
 sin
2

2
Product-to-Sum Formulas
sin u cos v = ½[sin(u + v) + sin(u – v)]
cos u sin v = ½[sin(u + v) – sin(u – v)]
cos u cos v = ½[cos(u + v) + cos(u – v)]
sin u sin v = ½[cos(u – v) – cos(u + v)]
Ex 6. Express sin3x sin5x as a sum of trig
functions.
Ex 7. Compute sin 15 cos 15.
Class Work
8. Write the product as a sum.
cos3x cos7 x
9. Compute: 3cos37.5 cos7.5
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
EX 8. Write sin 7 x  sin3x as a product.
Class Work
10. Write cos11x  cos5x as a product.
HW #2 p 548 1-7 odd, 15-23 odd, 27-31 odd,
35-57 odd.