# Few Important Points – Maths

i) sinnx, cosnx, secnx, cosecnx are periodic functions with period 2π and π according as n is odd or even.

ii) tannx, cotnx are periodic functions with period π whether n is even or odd.

iii) |sin x|, |cos x|, |tan x|, |cot x|, |sec x|, |cosec x| are periodic with period π.

iv) |sin x| + |cos x|, |tan x| + |cot x|, |sec x| + |cosec x| are periodic with period

v) sin-1 (sin x), cos-1 (cos x), cosec-1 (cosec x), sec-1 (sec x) are periodic with period whereas tan-1 (tan x) and cot-1 (cot x) are periodic with period π.

Example: Find the period of the function f(x) = e x – [x] + |cos πx| + |cos 2π x| + … + |cos n π x|

Solution: We observe that

Period of x – [x] is 1

Period of |cos π x| is π/π = 1

Period of |cos 2π x| is π/2π = 1/2

Period of |cos 3π x| is π/3π = 1/3

And so on.

Finally, Period of |cos nπ x| = π/nπ = 1/n

.·. Period of f (x) = LCM of  (1, ½. ⅓, …, 1/n) = 1.

EVEN AND ODD FUNCTIONS

EVEN FUNCTIONS: A function f (x) is said to be an even function if f (-x) = f (x) for all x.

ODD FUNCTIONS: A function f (x) is said to be an odd function if f (-x) = – f (x) for all x.

Hence, the values of x are

(-1 + √5)/2, (-1 – √5) /2, (-3 – √5)/2 and (-3 + √5)/2.

REMARK: Let f, g be two functions. Then

i) f is even, g is even => fog is an even function.

ii) f is odd, g is odd => fog is an odd function.

iii) f is even, g  is odd => fog is an even function

iv) f is odd, g  is even => fog is an even function.