Trigonometric Equation Solver

Solving Trigonometric Equations with Periodicity

An equation such as sin(2x + 1) = 0.5 does not have a single answer the way a linear equation does. Sine, cosine, and tangent are periodic functions, so any angle that produces the target value has infinitely many equivalents obtained by turning a full circle further. Solving fn(ax + b) = k therefore means finding not one number but an entire family of numbers, evenly spaced apart, that all satisfy the equation.

The first step is finding a reference angle θ₀ with the matching inverse function: arcsin(k) for sine, arccos(k) for cosine, or arctan(k) for tangent. Because sine and cosine only ever output values between −1 and 1, an equation with |k| greater than 1 has no solution at all, which the calculator flags immediately. Tangent has no such limit, since it ranges over every real number, but it repeats twice as often as sine and cosine — every π radians instead of every 2π — and is undefined wherever cos(θ) = 0.

Sine and cosine equations produce two distinct angle families per period, because each value is generally reached twice as the curve rises and falls: once at θ₀ and once at its mirror image (π − θ₀ for sine, −θ₀ for cosine). Tangent is one-to-one within each period, so it needs only a single family, built directly from θ₀.

Once the angle families are known, the last step undoes the ax + b substitution: since ax + b = θ, x = (θ − b) / a. Applying this to each family turns the angle’s own period into an x-period of 2π / |a| for sine and cosine, or π / |a| for tangent, giving the general solution shown above in the form x = x₀ + period·n, valid for every integer n.

When an interval is entered, the calculator narrows this infinite family down to the finitely many values of n that land inside it, listing each concrete solution in ascending order. This matches the way most textbooks phrase the question — “find all solutions in [0, 2π)”, for instance — and it is where the abstract general solution becomes a short, checkable list of numbers.