if cos (A + B) = cos A cos B - sin A sin B
then, cos 3 A = ?
Answers
Explanation:
Given:-
cos (A + B) = cos A cos B - sin A sin B
To find:-
The value of cos 3 A
Solution:-
Given that
cos (A + B) = cos A cos B - sin A sin B -----(1)
Now we have to find the value of Cos 3A
It can be written as
Cos (2A+A)
Put A = 2 A and B = A in the above formula (1) ,then
=>Cos 2A Cos A - Sin 2A sin A
And it can be written as
=>Cos (A+A) Cos A -Sin 2A Sin A
=>(Cos A CosA -Sin A Sin A ) CosA -Sin2A Sin A
=>(Cos^2 A - Sin^2 A ) Cos A - Sin 2A Sin A
=>(Cos^2×Cos A - Sin^2 A Cos A )-(Sin 2A SinA)
=>Cos^3 A - Sin^2ACosA - Sin 2A Sin A
We know that Sin^2 A + Cos^2 A = 1
=>Cos^3 A - (1-Cos^2 A)(Cos A) - Sin 2A Sin A
=> Cos^3 A -(Cos A -Cos^3 A) - Sin 2A Sin A
=>Cos^3 A -Cos A +Cos^3 A - Sin 2A Sin A
=>2Cos^3 A - Cos A - Sin 2A Sin A
We know that Sin 2A = 2 Sin A Cos A then
=>2Cos^3 A - Cos A -2 Sin A Cos A Sin A
=>2 Cos^3 A -Cos A -2 Sin^2 A Cos A
=>2 Cos^3 A -Cos A ( 1 +2 Sin^2 A)
=>2 Cos^3 A - Cos A (1+[2(1-Cos^2 A)]
=>2 Cos^3 A - Cos A[1+ (2 - 2Cos^2 A)
=>2 Cos^3 A - Cos A[1+2 - 2Cos^2 A)
=>2 Cos^3 A - Cos A[3 - 2Cos^2 A)
=>2 Cos^3 A - 3 Cos A +Cos A×2Cos^2 A
=>2 Cos^3 A - 3 Cos A + 2Cos^3 A
=>4 Cos^3 A - 3 Cos A
Answer:-
Cos 3A = 4 Cos^3 A - 3 Cos A
Used formulae:-
- cos (A + B) = cos A cos B - sin A sin B
- Sin^2 A + Cos^2 A = 1
- Sin 2A = 2 Sin A Cos A
Answer:
if you find it helpful then please mark me as a brainlist..
