Advanced trigonometric equations and inequalities

Let t = sin(x) and solve the quadratic. Reject t values outside [−1; 1]. For each remaining value, find all valid x; the number of t values is not the number of x solutions.

Equal tangent values repeat every 180°. Write the general inner-angle solution, rearrange for x and select roots in the interval. Check both endpoints, then add the solutions.

The tangent function has a period of 180°, so its value repeats after every half-turn.

1. Treat the given angle as one solution of the equation.
2. Add any integer multiple of 180° to it.
3. Use k to represent any integer.

General solution: x = α + 180°k, where k ∈ ℤ and α is the angle given in the question.

First find the angles where sine equals the right-hand side. Use the unit circle or sine graph to identify intervals satisfying the inequality. Include valid boundary points because the inequality is non-strict.

A product is zero when at least one factor is zero. Solve both equations separately and select roots in the interval. Count any shared root only once.

Express the double-angle sine as a product. Move all terms to one side and factor out the common factor. Do not divide by it, as that would lose roots. Select the largest valid root from both solution families.