Question

Which of the following is a solution to 2sin² x sin x - 1 = 0?

1) sin x = 1/2
2) sin x = -1/2
3) sin x = 1
4) sin x = -1

Answer 1

The correct solutions to the equation 2sin² x sin x - 1 = 0 are sin x = -1/2 and sin x = 1, which correspond to option 2 and option 3 respectively.

Step-by-step explanation:

The given equation is 2sin² x sin x - 1 = 0. To solve this trigonometric equation, we can substitute sin² x with (1 - cos² x) based on the Pythagorean identity. However, it seems more straightforward to look for factors of the quadratic in sin x. Let's set u = sin x to simplify the expression. The equation then becomes 2u² u - 1 = 0, which is a quadratic equation in u. Solving this quadratic equation yields solutions for u that correspond to possible values of sin x.

Factoring the quadratic, we get (2u + 1)(u - 1) = 0. This gives us the solutions: u = -1/2 and u = 1. Translating these back into terms of sin x, we have sin x = -1/2 and sin x = 1. Therefore, the possible solutions to the original equation are:

1. sin x = 1/2
2. sin x = -1/2
3. sin x = 1
4. sin x = -1

Comparing with the given options, option 2 (sin x = -1/2) and option 3 (sin x = 1) are the correct answers.