Question

Find the period for the function: 2 sin ($\frac{\pi}{4}$ + x) cos x

Answer 1

we know, if T is the period of function y = f(x) and R is the period of function, y = g(x) then period of y = f(x) ± g(x) will be LCM{T,R}

and also, if T is the period of function, y = f(x) then, T/|a| is the period of function, y = f(ax ± b)


now, f(x) = 2sin(π/4 + x)cosx

use formula, 2sinA.cosB = sin(A + B) + sin(A - B)

so, f(x) = sin(π/4 + x + x) + sin(π/4 + x - x)

f(x) = sin(2x + π/4) + sin(π/4)

f(x) = sin(2x + π/4) + 1/√2

we know, period of sinx = 2π

so, period of sin(2x + π/4) = 2π/2 = π

hence, period of function, f(x) = sin(2x + π/4) + 1/√2 is π

therefore, period of 2sin(π/4 + x)cosx is π