4.3 The multiple - angle (Double &
Half Angle) Formulas
With the help of the sum and difference (compound
angle formulas studied in the previous article, we will express
the trigonometric functions of angle
in terms of /2 ).
For any angle
(1) \ sin 2
= 2 sin cos
then
\ sin
= 2 sin ( /2) cos ( /2)
(2) \ cos 2
= cos2 -
sin2
\ cos
= cos2(
/2) - sin2(
/2)
\ cos 2
= 2 cos2
-1
\ cos
= 2 cos2(
/2) - 1 and
\ cos 2
= 1 - 2 sin2
\ cos
= 1 - 2 sin2 ( /2)
From the above formulas, we derive the following
formulas
![](r1.gif)
EXAMPLE 1 Find the value of
cos 150, using the ratios of 300only.
Solution:
![](r2.gif)
EXAMPLE 2 Find the exact value
for sin 1050using the half- angle identity.
Solution :
![](r3.gif)
![](r4.gif)
EXAMPLE 3 Find the exact value
of sin (292.50 ) using the half angle formulas.
Solution : /2
= 292.50 is in Quad. IV in which sine ratio is negative.
![](r5.gif)
EXAMPLE 4
Find the values of sine and cosine of
. Given that sin =
5 / 13 , is in Quad
II
Solution:
![](r7.gif)
EXAMPLE 5 Find the exact value
for cos 1650 using half- angle identity.
Solution:
![](r8.gif)
EXAMPLE 6 Find sin 22.50,
cos 22.50 and tan 22. 5 0using the half angle
formulas.
Solution:
![](r10.gif)
![](r11.gif)
EXAMPLE 7
in terms of sin x and cos x
Solution:
![](r13.gif)
EXAMPLE 8 Prove that
Solution
![](r15.gif)
EXAMPLE 9
![](r16.gif)
Solution:
![](r17.gif)
EXAMPLE 10
![](r18.gif)
Solution:
![](r19.gif)
EXAMPLE 11
![](r20.gif)
Solution:
![](r21.gif)
EXAMPLE 12
![](r22.gif)
Solution:
![](r23.gif)
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