Question

Find the periods for the given 1-5 functions
1. $\cos (3x+5)+7$
2. $\tan 5x$
3. $\cos \left(\frac{4x+9}{5}\right)$
4. $\left|\sin x\right|$
5. $\tan (x+4x+9x+....+n^{2x})$ (n any positive integer)

Answer 1

$\left(1\right)\cos (3x+5)+7$

period $=\frac{2\pi }{3}$

$\left(2\right)\tan 5x$

$period=\frac{\pi }{5}$

$\left(3\right)\cos (\frac{44}{5}+\frac{9}{5})$

$period=\frac{2\pi }{4}=\frac{5\pi }{2}$

$\left(4\right)\left|\sin \right|=\pi$

$\left(5\right)\tan (x+4x+9x+......+x^{2x})$

$x+4x+9x+......$

$x(1+2^2+3^2+4^2+......x^2)$

$x\left(\frac{x(x+1)(2x+1)}{6}\right)$

$\tan \left(\frac{x(x+1)(2x+1)}{6}x\right)$

period$=\frac{6\pi }{x(x+1)(2x+1)}$