What is a general solution?
It is a single expression, using an integer n, that captures every solution of a trig equation at once — rather than listing solutions inside a fixed domain.
// maths › Trigonometric Equations
General Solution
Write the general solution family for sin x = sinα, cos x = cosα or tan x = tanα, in degrees or radians, with the first few concrete solutions listed.
sin: x=α+360°n or 180°−α+360°n; cos: x=±α+360°n; tan: x=α+180°n
Angle unit
Frequently asked questions
What are the three patterns?
For sin x = sin α: x = α + 360°n or x = 180° − α + 360°n. For cos x = cos α: x = ±α + 360°n. For tan x = tan α: x = α + 180°n. In radians replace 360° with 2π and 180° with π.
Can you show a worked example?
For sin x = sin 30°, the general solution is x = 30° + 360°n or x = 150° + 360°n. Putting n = 0, 1 gives 30°, 150°, 390°, 510°… In radians α = π/6 and the period is 2π; switch the toggle to see it.
How do I get the in-range solutions back?
Substitute the integer values of n that land inside the required domain. Each n gives one concrete solution, so a domain of one turn usually picks out two of them.
Where is this used?
Anywhere a periodic condition recurs indefinitely — repeated alignment of rotating parts, recurring tide or daylight conditions, and resonance timing.