TRIGONOMETRIC FUNCTIONS · Part 21

n π + y , where n ∈ Z Hence x = ( n + ) π + (– ) n + y or x = n π +(– ) n y , where n ∈ Z . Combining these two results, we get x = n π + (– ) n y , where n ∈ Z . Theorem For any real numbers x and y , cos x = cos y , implies x = n π ± y , where n ∈ Z Proof If cos x = cos y , then cos x – cos y = i.e., – sin sin = Thus sin = or sin = Therefore = n π or = n π , where n ∈ Z i.e. x = n π – y or x = n π + y , where n ∈ Z Hence x = n π ± y , where n ∈ Z Theorem Prove that if x and y are not odd mulitple of π , then tan x = tan y implies x = n π + y , where n ∈ Z MATHEMATICS Proof If tan x = tan y , then tan x – tan y = or sin cos cos sin cos cos = which gives sin ( x – y ) = (Why?) Therefore x – y = n π , i.e., x = n π + y , where n ∈ Z Example Find the solution of sin x = – Solution We have sin x = – = π π π sin sin π sin Hence sin x = π sin , which gives π π ) + − , where n ∈ Z .