Explanation:
To solve the equation:
6(sin(x))^2 + 7cos(x) - 3 = 0
We can use the identity:
sin^2(x) + cos^2(x) = 1
Rearranging the equation, we get:
6(1-cos^2(x)) + 7cos(x) - 3 = 0
Expanding and rearranging, we get:
6cos^2(x) + 7cos(x) - 9 = 0
This is now a quadratic equation in terms of cos(x).
Using the quadratic formula, we get:
cos(x) = [-7 ± √(7^2 - 4(6)(-9))]/(2(6))
cos(x) = [-7 ± 13]/12
cos(x) = 1/2 or -3/2
Now we use the inverse cosine function to find x for each solution for cos(x).
When cos(x) = 1/2, we get:
x = π/3 + 2πk or x = 5π/3 + 2πk
When cos(x) = -3/2, we get:
there are no solutions for this case.
Therefore, the general solution to the equation is:
x = π/3 + 2πk or x = 5π/3 + 2πk where k is an integer.