Some nice manipulations:
1)
Why ? = cos² sin² 2sin cos
= 1 2sincos
= 1 sin²


2)
Why ?

=

3) cos Asin A =
Why !
cos A sin A =
=
=

Expressing sin in terms of sin A:

=
=
The ambiguity of sign is removed from following figure.



Dumb Question: Where does this diagram come from ?
Ans:-
So, if is in I ^st or II^nd quadrent
then is + ve.
So, A/2 lie from 2_n for to be + ve
Similarily for

Illustration 7:
If cos 25⁰ + sin 25 ⁰ = P , then find value of cos 50⁰ in trems of P ?
Ans:- cos50⁰ = cos² 25 ⁰ - sin² 25⁰
= (cos 25⁰ + sin25⁰)(cos 25⁰ - sin25⁰)
P(cos25⁰ - sin25⁰)
Also, (cos25⁰ - sin25⁰)² + (cos25⁰ - sin25⁰)² = 1 + 1
cos25⁰ - sin25⁰ = +
(+ve sign as cos 25⁰ > sin 25⁰)
cos50⁰ =

The qreatest and least valume of expression (a sin + b cos)

- a sin + b cos


Why?
Let a = r cos
b = r sin so that r =
So, a sin + b cos = r (sin cos + cossin )
= r sin ( + )
Now sin ( + has minimum and maximum value as + 1 and - 1 esppectively.
So, - r rsin ( + ) r
So, - a sin + b cos

Illustration 8:
Find the minimum and maximum value of
6 sin x cos x + 4cos² 2x ?
Ans:- 6 sin x cos x + 4cos² 2x = (2sin x cos x ) + 4cos 2x
= 3 sin 2x + 4cos 2x .
- 3sin 2x + 4cos2x
=> Minimum value of 6sin x cos x + 4 cos 2x is - 5
and maximum value is 5.

Sum of sine and cosine senies when angles are in AP .
(1) sin + sin =
Why ?

2sin
2sin
2sin
By adding these n lines we have.
2sin
= 2sin
S =

(2)
=

Illustration 9:
Find sum of sin -............+ 0 n terms. Ans:- Now, sin
sin
sin
........................................................
Hence the series is
sin ............
=
=