Download Multiple-Angle Formulas (Sec. 7.4)

Multiple-Angle Formulas (Sec. 7.4)
Feature
Details / Examples
Multiple angle formulas
Formulas for trigonometric functions of multiples of an angle
Double-angle Formulas
Formulas for trigonometric functions of twice an angle in terms
of functions of the basic angle.
1. sin 2u = 2 sin u cos u
2. (a) cos 2u = cos2 u – sin2 u
(b) cos 2u = 1 – 2 sin2 u
(c) cos 2u = 2 cos2 u – 1
3. tan 2u = 2 tan u / (1 – tan2 u)
Example 1: Using doubleangle formulas
If sin α = 4/5 and α is an acute angle, find the exact values of
sin 2α and cos 2α.
Half-Angle Identities
1. sin 2 u =
1 − cos 2u
2
2. cos 2 u =
1 + cos 2u
2
3. tan2 u =
1 − cos 2u
1 + cos 2u
€
€
Example 2: Using half- €
angle identities to verify
an identity
Verify the identity sin2 x cos2 x = 1/8(1–cos 4x) .
1
Half-Angle Formulas
v
1 − cos v
sin  = ±
2 
2
v
1 + cos v
cos  = ±
2 
2
€
€
Half-Angle Formulas vs.
Half-Angle Identities
€
Half-Angle Formulas for
Tangent
Example 3
€
v
1 − cos v
tan  = ±
2
1 + cos v
Why are some of these called “formulas” while others are
called “identities”? Obviously the “formulas” aren’t identities,
right, Mr. Byrd? __________.
1. tan
v 1 − cos v
=
2
sin v
2. tan
v
sin v
=
2 1 + cos v
€
DAB, April 2011
2