# Trigonometry

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Find the general solution of the following equation.
sin (5x) = cos (3x)
sin (6x) sin (x) = cos (3x) cos (4x)

• Trigonometry -

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

• Trigonometry -

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.

• Trigonometry -

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

• Trigonometry -

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.

• Trigonometry (shortcut) -

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.

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