12sinx + 5cosx +13=0

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To solve this equation, we can rewrite it as:

12sin(x) + 5cos(x) = -13

Now, we can square both sides of the equation to eliminate the trigonometric functions:

(12sin(x) + 5cos(x))^2 = (-13)^2

Expanding the left side using the trigonometric identity sin^2(x) + cos^2(x) = 1 and the angle addition formula for sine and cosine (sin(a + b) = sin(a)cos(b) + cos(a)sin(b)), we get:

144sin^2(x) + 120sin(x)cos(x) + 25cos^2(x) = 169

Using the Pythagorean identity (sin^2(x) + cos^2(x) = 1), we can rewrite the equation as:

144(1 - cos^2(x)) + 120sin(x)cos(x) + 25cos^2(x) = 169

Expanding and simplifying the equation, we get a quadratic equation in terms of cos(x):

169 - 144cos^2(x) + 120sin(x)cos(x) + 25cos^2(x) = 169
-119cos^2(x) + 120sin(x)cos(x) = 0

Dividing both sides by cos(x) (assuming cos(x) is not equal to zero), we get:

-119cos(x) + 120sin(x) = 0

Dividing by cos(x), the equation becomes:

-119 + 120tan(x) = 0

Rearranging, we find:

tan(x) = 119/120

Which gives us the solution for x.