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:
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.