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
= cos^{2} 
sin^{2}
\ cos
= cos^{2}(
/2)  sin^{2}(
/2)
\ cos ^{2}
= 2 cos^{2}
1
\ cos
= 2 cos^{2}(
/2)  1 and
\ cos 2
= 1  2 sin^{2}
\ cos
= 1  2 sin^{2} (/2)
From the above formulas, we derive the following
formulas
EXAMPLE 1 Find the value of
cos 15^{0}, using the ratios of 30^{0}only.
Solution:
EXAMPLE 2 Find the exact value
for sin 105^{0}using the half angle identity.
Solution :
EXAMPLE 3 Find the exact value
of sin (292.5^{0} ) using the half angle formulas.
Solution : /2
= 292.5^{0} is in Quad. IV in which sine ratio is negative.
EXAMPLE 4
Find the values of sine and cosine of
. Given that sin =
5 / 13 , is in Quad
II
Solution:
EXAMPLE 5 Find the exact value
for cos 1650 using half angle identity.
Solution:
EXAMPLE 6 Find sin 22.5^{0},
cos 22.5^{0} and tan 22. 5 ^{0}using the half angle
formulas.
Solution:
EXAMPLE 7
in terms of sin x and cos x
Solution:
EXAMPLE 8 Prove that
Solution
EXAMPLE 9
Solution:
EXAMPLE 10
Solution:
EXAMPLE 11
Solution:
EXAMPLE 12
Solution:
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