Trigonometric Equation Solver
Solve trigonometric equations like sin x = 1/2, cos 2x = −√3/2 or tan x = 1: the principal value, the general solution and every solution from 0° to 360°.
About the Trigonometric Equation Solver
An equation like sin x = 1/2 has infinitely many solutions, because sine repeats. Find one angle α (the principal value, from sin⁻¹), then use the general solution: for sine, x = nπ + (−1)ⁿα; for cosine, x = 2nπ ± α; for tangent, x = nπ + α, where n is any whole number.
When a question gives an interval such as 0 ≤ x < 2π, list the values of n that put x inside it.
The formula
sin x = sin α ⟹ x = nπ + (−1)ⁿα; cos x = cos α ⟹ x = 2nπ ± α; tan x = tan α ⟹ x = nπ + α
Worked example
sin x = 1/2
- α = π/6
- x = nπ + (−1)ⁿ π/6
- In [0°, 360°): 30° and 150°
Questions people ask
Why is the principal value of cos⁻¹ between 0° and 180°?
Each inverse function picks one angle from a range where the function takes every value once: [−90°, 90°] for sin⁻¹ and tan⁻¹, [0°, 180°] for cos⁻¹.
What about sin 2x = k?
Set m = 2: solve 2x = nπ + (−1)ⁿα, then divide by 2. There are twice as many solutions in [0°, 360°).