such equations. We have already learnt that the values of sin x and cos x repeat after an interval of π and the values of tan x repeat after an interval of π . The solutions of a trigonometric equation for which ≤ x < 2π are called principal solutions . The expression involving integer ‘ n ’ which gives all solutions of a trigonometric equation is called the general solution .
We shall use ‘ Z ’ to denote the set of integers. The following examples will be helpful in solving trigonometric equations: Example Find the principal solutions of the equation sin x = . Solution We know that, π sin and π π π sin sin π sin Therefore, principal solutions are π x = and π . Example Find the principal solutions of the equation tan x = − .
Solution We know that, π tan . Thus, π π tan π – = – tan = – and π π tan π tan = − = − Thus π π tan tan = − Therefore, principal solutions are π and π . We will now find the general solutions of trigonometric equations. We have already TRIGONOMETRIC FUNCTIONS seen that: sin x = gives x = n π , where n ∈ Z cos x = gives x = ( n + ) π , where n ∈ Z .
We shall now prove the following results: Theorem For any real numbers x and y , sin x = sin y implies x = n π + (– ) n y , where n ∈ Z Proof If sin x = sin y , then sin x – sin y = or 2cos sin = which gives cos = or sin = Therefore = ( n + ) π or = n π , where n ∈ Z i.e. x = ( n + ) π – y or x =