Find the general solution of the following equation.

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

Thanks for your help :)

Those are two different equations.

Which one do you want to solve?

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.

No one helped me ?!

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

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.

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.

To find the general solution of the equation sin(5x) = cos(3x), we can use the trigonometric identity cos(x) = sin(π/2 - x).

Let's rewrite the equation using this identity:

sin(5x) = sin(π/2 - 3x)

Now, we have two cases to consider:

Case 1: sin(5x) = sin(π/2 - 3x)

In this case, the equation becomes:

5x = π/2 - 3x

Now, solve for x:

5x + 3x = π/2
8x = π/2
x = π/16 + nπ/4 (where n is an integer)

So, the general solution for this case is:

x = π/16 + nπ/4 (where n is an integer)

Case 2: sin(5x) = sin(3x - π/2)

In this case, the equation becomes:

5x = 3x - π/2

Now, solve for x:

5x - 3x = -π/2
2x = -π/2
x = -π/4 + nπ/2 (where n is an integer)

So, the general solution for this case is:

x = -π/4 + nπ/2 (where n is an integer)

Now, let's move on to solving the second equation: sin(6x) sin(x) = cos(3x) cos(4x).

Using the identity sin(x) = cos(π/2 - x) and cos(x) = sin(π/2 - x), we can rewrite the equation as:

cos(π/2 - 6x) cos(π/2 - x) = cos(3x) cos(4x)

Applying the identity cos(x) = sin(π/2 - x), we get:

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

Now, we have two cases to consider:

Case 1: sin(6x) = sin(3x) and sin(x) = sin(4x)

In this case, the equations become:

6x = 3x + 2kπ (1)
x = 4x + jπ (2)

Solving equation (1) for x:

6x - 3x = 2kπ
3x = 2kπ
x = 2kπ/3 (where k is an integer)

Substituting this value of x into equation (2):

2kπ/3 = 4(2kπ/3) + jπ
2kπ/3 = 8kπ/3 + jπ
-6kπ/3 = jπ - 2kπ
-2k = j - 6

Hence, j = -2k + 6

So, the general solution for this case is:

x = 2kπ/3, j = -2k + 6 (where k is an integer)

Case 2: sin(6x) = sin(3x) and sin(x) = -sin(4x)

In this case, the equations become:

6x = 3x + 2kπ (1)
x = -4x + jπ (2)

Solving equation (1) for x:

6x - 3x = 2kπ
3x = 2kπ
x = 2kπ/3 (where k is an integer)

Substituting this value of x into equation (2):

2kπ/3 = -4(2kπ/3) + jπ
2kπ/3 = -8kπ/3 + jπ
10kπ/3 = jπ
j = 10k/3

So, the general solution for this case is:

x = 2kπ/3, j = 10k/3 (where k is an integer)

Therefore, the general solution for the second equation is:

x = 2kπ/3, j = -2k + 6 (where k is an integer)

I hope this explanation helps you understand the steps to find the general solution for these equations!