


TrigonometryGeneral Solutions of a Trig EquationFrom 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
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
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
Example Let us find the general solutions of 2cos x = 1. The equation is equivalent to Now . Hence the general solution is
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