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Section 6.2
Double-Angle and Half-Angle Formulas
The following, most useful, basic identities follow from the
addition formulas.
Double-Angle Formulas
Half-Angle Formulas
Note: In the half-angle formulas the ± symbol is intended to
mean either positive or negative but not both, and the sign before
θ
the radical is determined by the quadrant in which the angle
2
terminates.
Example 1: Suppose sin θ = −
4
3π
and π < θ < .
5
2
a. Find sin ( 2θ ) .
θ
b. Find cos   .
2
Section 6.2 – Double-Angle and Half-Angle Formulas
1
c. Find sin d. Find tan When calculating trigonometric functions of multiples of ,
you have the choice of using an addition formula or using a
half-angle formula.
When calculating trigonometric functions of multiples of
you have only one choice: a half-angle formula.
It is not possible to write
special angles ,
,
as a sum or difference of our
, and !!!
π
Example 2: Calculate cos   .
8
Section 6.2 – Double-Angle and Half-Angle Formulas
2
 13π 
Example 3: Use the half-angle formula to calculate sin 
.
 12 
Example 4: Use the half angle formula to evaluate Section 6.2 – Double-Angle and Half-Angle Formulas
.
3
Example 5: Given =
√
. Where x is an acute angle, find
Section 6.2 – Double-Angle and Half-Angle Formulas
4