section 7.3 double-angle, half-angle and product-sum formulas
DESCRIPTION
Section 7.3 Double-Angle, Half-Angle and Product-Sum Formulas. Objectives: To understand and apply the double-angle formula. To understand and apply the half-angle formula. To understand and apply product-sum formula. Class Work. Evaluate csc780 Find the exact value of tan165. - PowerPoint PPT PresentationTRANSCRIPT
Section 7.3Double-Angle, Half-Angle and
Product-Sum FormulasObjectives:•To understand and apply the double-angle 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:
Formula for cosine:
Formula for tangent:
sin 2 2sin cosx x x
2 2cos 2 cos sinx x x
2
2 tantan 2
1 tan
xx
x
21 2sin x 22cos 1x
Ex 1. If and x is in quadrant II, find cos 2x, sin 2x, and tan 2x.
2cos
3x
Ex 2. If and x is in quadrant I, then find sin 2x, cos 2x, and tan 2x.
3sin
5x
Class Work
3. If and x is in quadrant II, find sin 2x, cos 2x, and tan 2x.
6cos
11x
Half-Angle Formulas1 cos
sin2 2
1 coscos2 2
1 costan2 sin
sin
1 cos
u u
u u
u u
uu
u
Ex 3. Find the exact value of sin 22.5.
Ex 4. Find tan8
Ex 5. Find if and u is in quad II. tan2
u 2sin
5u
Class Work
4. Find if and u is in quad IV.
sin , cos , tan2 2 2
u u uand
2cos
3u
Simplify the expression by using a double-angle or half-angle formula.
5.
6.
7.
2sin35 cos35
2 2cos sin2 2
1 cos4
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 as a sum of trig functions.
sin 3 sin 5x x
Ex 7. Compute sin 15 cos 15.
Class Work
8. Write the product as a sum.
9. Compute:
cos3 cos7x x
3cos37.5 cos7.5
Sum-to-Product Formulas
sin sin 2sin cos2 2
sin sin 2cos sin2 2
cos cos 2cos cos2 2
cos cos 2sin sin2 2
x y x yx y
x y x yx y
x y x yx y
x y x yx y
EX 8. Write as a product. sin 7 sin 3x x
Class Work
10. Write as a product. cos11 cos5x x
HW #2 p 548 1-7 odd, 15-23 odd, 27-31 odd, 35-57 odd.