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## General Solutions of a Trig Equation

From the following diagram we see that sin(π -θ) = sin θ and cos ( -θ) = cos θ. We use this to find the solutions of some trig equations.

Solve sin(x) = y for x. Case 1: -1≤y≤ 1, that is, the value of y is between -1 and 1, so there is a solution.

The set of all solutions to sin(x) = y is

x = sin-1(y) + 2 and x = −sin-1(y) + (2k+1)π,

where k can be any integer; that is, the solutions for x consist of sin-1(y) plus all even multiples of π, together with minus sin-1(y) plus all odd multiples of π

Case 2: -1 > y or y > 1 , that is, the value of y is too large or too small for a solution to be possible.

There are no solutions.

Solve cos(x) = y for x. Case 1: -1≤y≤ 1

The set of all solutions to cos(x) = y is

x = ± cos-1(y)+ 2 ,

where k can be any integer;

Case 2: -1 > y or y > 1

There are no solutions.

Solve tan(x) = y for x. The set of all solutions to tan(x) = y is

x = tan-1(y) + , where k can be any integer.

Example

Let us find the general solutions of  2cos x = 1.

The equation is equivalent to Now . Hence the general solution is

x = ±π/3 + 2, where k is any integer.