trignometric-ratios-and-identities-3

Trignometri Ratios multiples of an angle :
1) sin 2 = 2sin cos=
why ? sin (A +B) = sin A cos B + cos A sin B
Put A = B =
So, sin 2 = 2sin cos
=
=

2) cos2 = cos² -sin² = 1 - 2 sin²
= cos² - 1
=

3) tan 2 =
4) sin 3 = 3 sin - 4sin²
Why ?
sin3 = sin ( + 2 )
sin ( + 2) = sin cos²+ cos sin²
= sin ( 1 - 2 sin²) + cos (2sincos)
= sin - 2sin³ + 2sin (1 - sin²)
= 3 sin - 4 sin³

5) cos 3 = 4 cos³ - 3 cos

6) tan 3 =

Illustration 5:
sin x + sin y = a and cos x + cos y = b , show that
sin (x + y) = and tan (x - y)
Ans:- sin x + sin y = a
=> 2 sin = a ...................(1)
cos x + cos y = b
=> 2 cos = b ...................(2)
=> tan
sin (x + y) =
=
Squaring (1), (2) and adding .
4 cos² = a² +b²
or cos² =
sin² = 1 -
=
tan²
tan =


Illustration 6:-
Find the value of sin18⁰ ?
Ans:- Let 18⁰
So, 5 = 90⁰
=> 2 = 90⁰ - 3
Or, sin 2 = sin (90⁰ - 3)
=> sin 2 =cos3
=> 2sin cos= cos(4 cos² - 3)
=> 2sin = 4cos² - 3 ( cos 0)
= 1 - 4sin²
=> 4sin² + 2 sin - 1 = 0
So, sin =
=
But since sin > 0 we have sin =
So, sin 18 ^o =

Dumb Question:- Why cos is not equal to 0 ?
Ans: cos 90⁰ si O and cos O^o is 1
So, cos being a continous function.
cosi.e. cos 18⁰ would have some value between o and 1.