How to solve basic trigonometric equations?

Check the range and isolate the base angle

Solving basic trigonometric equations begins with checking whether a real solution exists. Because the sine and cosine functions never leave the range [-1, 1], an equation such as sin x = 1.5 has no solution. Tangent equations have no such restriction because the tangent function covers all real numbers.

Once you confirm a valid value, find the principal reference angle using the inverse function. For example, if sin x = 1/2, the principal angle in the first quadrant is π/6 radians (or 30 degrees).

Find both quadrant solutions and add the period

Trigonometric values repeat across quadrants. Sine is positive in Quadrants I and II, while cosine is positive in Quadrants I and IV. For sin x = 1/2, your base solutions are x = π/6 and x = π - π/6 = 5π/6.

Because the graph repeats endlessly, write the general solution by adding integer multiples of the period. Sine and cosine repeat every 2π radians, giving x = π/6 + 2πk and x = 5π/6 + 2πk, where k is an integer. Tangent repeats every π radians, so tan x = 1 yields x = π/4 + πk.

Restrict solutions to a specified interval

Many problems ask for solutions within a specific domain, such as [0, 2π). Substitute integer values for k (such as 0, 1, or -1) into your general formulas to collect every value that falls inside the required interval, discarding any that lie outside.