Find the general solution of the following equation.

Question

sin (5x) = cos (3x)
sin (6x) sin (x) = cos (3x) cos (4x)

Thanks for your help :)

Answer 1

Those are two different equations.
Which one do you want to solve?

Answer 2

I've tried the first one, and i found out that x=45n+11.25 or x=45-180n but i don't know if i am correct or not.
The second one I just cant do it.

Answer 3

No one helped me ?!
How poor I am ..........

Answer 4

I could write both equations in terms of sines and cosines of x, but they become an incredible mess. There may be an elegant solution to either, but don't see it.

Answer 5

sin (5x) = cos (3x)
That relationship is true if the argument of the sin term is y and the argument of the cos term is pi/2 - y.
Let
5x = y, and
3x = pi/2 -y
Now solve for y.
15x = 3y
15x = 5 pi/2 -5y

0 = 8y - 5 pi/2
y = 5 pi/16 = 56.25 degrees is an answer

There may be another answer as well, since the
sin a = cos b
relationsip is also true whenever the argument of the sin term is y and the argument of the cos term is 3 pi/2 + y.

So let

5x = y
3x = 3 pi/2 + y
15x = 3y
15x = 15 pi/2 + 5 y
2y = -15 pi/2
So y = -15 pi/4 = -675 degrees, or -315 degrees, or +45 degrees is another answer.