Posted by **Tommy** on Tuesday, July 22, 2008 at 7:35am.

Find the general solution of the following equation.

sin (5x) = cos (3x)

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

Thanks for your help :)

- Trigonometry -
**drwls**, Tuesday, July 22, 2008 at 7:42am
Those are two different equations.

Which one do you want to solve?

- Trigonometry -
**Tommy**, Tuesday, July 22, 2008 at 7:47am
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 -
**Tommy**, Tuesday, July 22, 2008 at 10:38am
No one helped me ?!

How poor I am ..........

- Trigonometry -
**drwls**, Tuesday, July 22, 2008 at 11:02am
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) -
**drwls**, Tuesday, July 22, 2008 at 3:10pm
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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