Download SECTION 10-3 Double Angle and Half

SECTION 10-3
Double Angle and Half-Angle Formulas
TRIGONOMETRIC FUNCTIONS IN REALLIFE


Review how trigonometric functions are use in
science and engineering to study light and sound
waves.
Review graphs on p. 380.
SIN 2X AND SIN ½X


If you know the value of sin α, you do NOT double
it to fine sin 2α. Nor do you halve it to find sin
½α.
Complete activity on p. 380 – notice that the
values are not the same for each graph.
DOUBLE-ANGLE FORMULAS
The following double-angle formulas are special
cases of the formulas for sin (α + β), cos (α + β),
and tan (α + β). If we let β = α in these formulas
we obtain the following formulas.
 sin (α + β) = sin α cos β + cos α sin β
 sin (α + α) = sin α cos α + cos α sin α
 sin 2α = 2 sin α cos α

DOUBLE-ANGLE FORMULAS
cos (α + β) = cos α cos β- sin α sin β
 cos (α + α) = cos α cos α – sin α sin α

cos 2  cos 2   sin 2 

Using the fact thatsin 2   cos 2   1we can obtain
alternative formulas for cos 2α:
cos 2  1  2 sin 2 
cos 2  2 cos 2   1
DOUBLE-ANGLE FORMULAS

To express tan 2α in terms of tan α, we again let
β = α.
tan   tan 
tan     
1  tan  tan 
tan   tan 
tan     
1  tan  tan 
2 tan 
tan 2 
1  tan 2 
HALF-ANGLE FORMULAS

To obtain the sine and cosine half-angle formulas, we
use formulas (8b) and (8c), replacing α with 
2
2
cos 2  1  2 sin 2 
 x
2 x
cos 2   1  2 sin
2
2
x
cos x  1  2 sin 2
2
x
2 sin 2  1  cos x
2
sin
x
1  cos x

2
2
cos 2  2 cos   1
 x
2 x
cos 2   2 cos
1
2
2
x
cos x  2 cos 2  1
2
x
2 cos 2  1  cos x
2
cos
x
1  cos x

2
2
HALF-ANGLE FORMULAS

When you use the half-angle formulas choose + or
– depending on the quadrant in which x lies.
2
HALF-ANGLE FORMULA FOR TAN
x
 To derive a formula for tan
, divide equation (10) by
2
equation (11):
tan

The following formulas which can be derived by
simplifying the radical expression in formula (12) may
be more useful.
x
sin x
tan 
2 1  cos x

x
1  cos x

2
1  cos x
x 1  cos x
tan 
2
sin x
Notice that these formulas don’t need the ambiguous
sign ±.
DOUBLE-ANGLE AND HALF-ANGLE
FORMULAS
EXAMPLE

Simplify
 
 
sin    cos 2  
2
2
2
2 tan 157.7
1  tan 2 157.7
EXAMPLE

Use a half-angle formula to evaluate each
expression.
sin (-67.5°)
cos

4
EXAMPLE
1
 If sin A =
3

A
and     find cos 2A and cos
2
2