How do I solve a trig equation over a domain?
Take the principal value from the inverse function, then use the function's symmetry to find the partner solutions, keeping only those inside the stated domain (usually one full turn).
// maths › Trigonometric Equations
Solving Trig Equations
Solve sin x = k, cos x = k or tan x = k over a chosen domain (degrees or radians), listing every solution in range with the working shown.
sin x = k → x = sin⁻¹k and its symmetric partners, within the domain
Angle unit
Frequently asked questions
Can you show a worked example?
For sin x = 0.5 over [0°,360°): sin⁻¹(0.5) = 30°, and the supplement 180°−30° = 150°, so x = 30° or 150°. In radians the domain is [0,2π) and the answers are π/6 ≈ 0.5236 and 5π/6 ≈ 2.6180 rad.
Why can an equation have no solution?
sin x and cos x always lie between −1 and 1, so sin x = 2 has no solution. tan x can equal any real number, so a tangent equation always has solutions.
How many solutions are there in one turn?
Usually two for sine and cosine (a value and its symmetric partner) and one for tangent per 180°, but at special values such as sin x = 1 the two partners coincide into a single solution.
Where is this used?
Finding when an oscillation reaches a given value — times of a given tide height, phase angles in AC circuits, or launch angles that hit a target.