A Note π is one such value of x for which sin x = − . One may take any other value of x for which sin x = − . The solutions obtained will be the same although these may apparently look different. Example Solve cos x = .
Solution We have, π cos cos x = Therefore π π ± , where n ∈ Z . Example Solve π tan cot = − Solution We have, π tan cot = − π π tan TRIGONOMETRIC FUNCTIONS or π tan2 tan Therefore π π , where n ∈ Z or π π , where n ∈ Z . Example Solve sin x – sin x + sin x = . Solution The equation can be written as sin sin sin or sin cos sin i.e.
sin ( cos x − Therefore sin x = or cos x = i.e. π sin4 or cos cos Hence π π or π x n ± , where n ∈ Z i.e. π π or π ± , where n ∈ Z . Example Solve cos x + sin x = Solution The equation can be written as sin sin or sin sin or (2sin ) (sin ) Hence sin x = or sin x = But sin x = is not possible (Why?) Therefore sin x = π sin .
MATHEMATICS Hence, the solution is given by π π ) + − , where n ∈ Z . EXERCISE . Find the principal and general solutions of the following equations: . tan x = .
sec x = . cot x = − . cosec x = – Find the general solution for each of the following equations: . cos x = cos x .
cos x + cos x – cos x = . sin