Find all solutions to a trigonometric equation on the interval [0, 2π). Handles simple equations like A·f(x) = B and quadratic-in-trig equations like 2sin²x − sin x − 1 = 0. Get exact π-fraction answers, step-by-step work, and unit circle references.
Equation Setup
Solutions on [0, 2π)
Once you know the principal value θ = arcsin/arccos/arctan(v), use symmetry to find the partner angle:
sin(x) = v → x = θ, π − θ
cos(x) = v → x = θ, 2π − θ
tan(x) = v → x = θ, θ + π
Adjust any negative angle by adding 2π. Keep only values in [0, 2π). Sort results in ascending order.
Treat the trig function as a single variable u, solve the quadratic, then find angles:
2sin²x − sinx − 1 = 0
Let u = sin(x):
2u² − u − 1 = 0 → (2u + 1)(u − 1) = 0
Roots: u = −1/2 and u = 1. Each valid root (|u| ≤ 1 for sin/cos) gives its own set of angles on [0, 2π). Discard roots where |u| > 1 — no real solution exists.
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