The solutions to the trigonometric equation 10sin²(x) - 3sin(x) - 1 = 0 are x = 30 degrees and x = 150 degrees. The substitution u = sin(x) transforms the equation into (2u - 1)(5u + 1) = 0, leading to valid solutions for sin(x) of 1/2 in the first and second quadrants.
To solve the trigonometric equation 10sin²(x) - 3sin(x) - 1 = 0, we can use the substitution u = sin(x). This transforms the equation into a quadratic equation in terms of u: 10u² - 3u - 1 = 0.
Now, we can factor the quadratic equation as (2u - 1)(5u + 1) = 0. This implies that either 2u - 1 = 0 or 5u + 1 = 0. Solving each factor separately gives us:
For 2u - 1 = 0, we find u = 1/2.
For 5u + 1 = 0, we find u = -1/5.
Now, recall that u = sin(x). Therefore, sin(x) = 1/2 or sin(x) = -1/5.
Now, let's express the solutions for x:
When sin(x) = 1/2: This occurs for x in the first and second quadrants. In these quadrants, x can be 30 degrees or 150 degrees.
When sin(x) = -1/5: This solution is not valid, as the sine function's range is [-1, 1] and cannot equal -1/5.
Therefore, the valid solutions for the equation 10sin²(x) - 3sin(x) - 1 = 0 are x = 30 degrees and x = 150 degrees.