Explanation:
I assume you mean the equation sin(x)tan(x) + cos(x) = 3.
Let's begin by rearranging the equation:
sin(x)tan(x) = 3 - cos(x)
Now, we can use the identity tan(x) = sin(x) / cos(x) to rewrite the left-hand side:
sin(x) * sin(x) / cos(x) = 3 - cos(x)
sin^2(x) = (3 - cos(x)) * cos(x)
Expanding the right-hand side, we get:
sin^2(x) = 3cos(x) - cos^2(x)
Moving all the terms to the left-hand side, we obtain a quadratic equation in terms of cos(x):
cos^2(x) - 3cos(x) + sin^2(x) = 0
Using the identity sin^2(x) + cos^2(x) = 1, we can substitute sin^2(x) = 1 - cos^2(x) to get:
cos^2(x) - 3cos(x) + 1 = 0
Solving for cos(x) using the quadratic formula, we get:
cos(x) = (3 ± sqrt(5)) / 2
Now, we can use the inverse cosine function to find the corresponding values of x:
x = arccos((3 + sqrt(5)) / 2) ≈ 20.10° or x = arccos((3 - sqrt(5)) / 2) ≈ 159.90°
Therefore, the solutions to the equation sin(x)tan(x) + cos(x) = 3 for 0° < x < 360° are approximately x = 20.10° and x = 159.90°.