= cos π = − sin π = cos π = sin π = – cos π = sin π = Now, if we take one complete revolution from the point P, we again come back to same point P. Thus, we also observe that if x increases (or decreases) by any integral multiple of π , the values of sine and cosine functions do not change. Thus, sin ( n π + x ) = sin x , n ∈ Z , cos ( n π + x ) = cos x , n ∈ Z Further, sin x = , if x = , ± π, ± π , ± π , ..., i.e., when x is an integral multiple of π and cos x = , if x = ± π , ± π , ± π , ... i.e., cos x vanishes when x is an odd multiple of π .
Thus sin x = implies x = n π, π, π, π, π, where n is any integer cos x = implies x = ( 2n + ) π , where n is any integer. We now define other trigonometric functions in terms of sine and cosine functions: cosec x = sin x , x ≠ n π, where n is any integer. sec x = cos x , x ≠ ( n + ) π , where n is any integer. tan x = sin cos x , x ≠ ( n + ) π , where n is any integer.
cot x = cos sin x , x ≠ n π , where n is any integer. TRIGONOMETRIC FUNCTIONS not defined not defined We have shown that for all real x , sin x + cos x = It follows that + tan x = sec x (why?) + cot x = cosec x (why?) In earlier classes, we have