Question

Which function is odd? Check all that apply.

Answer 1

$The\ function\ is\ odd\ if\ f(-x)=-f(x)\\\\We\ know:\\\sin x;\ \tan x\ and\ \cot x\ are\ odds.\\\cos x\ is\ even\ /\cos(-x)=\cos x/\\--------------------\\\\A.\ y=\sec x\\\\\sec(-x)=\dfrac{1}{\cos(-x)}=\dfrac{1}{\cos x}=\sec x-\fbox {NO}\\\\B.\ y=\sin x-\fbox{YES}\\\\C.\ y=\cot x\\\\\cot(-x)=\dfrac{\cos(-x)}{\sin(-x)}=\dfrac{\cos x}{-\sin x}=-\dfrac{\cos x}{\sin x}=-\cot x-\fbox{YES}\\\\D.\ y=\csc x\\\\\csc(-x)=\dfrac{1}{\sin(-x)}=\dfrac{1}{-\sin x}=-\dfrac{1}{\sin x}=-\csc x-\fbox{YES}$

Answer 2

An odd function is  a function which when x is substituted with -x, the function is equal to the negative counterpart of the original function. we use pi as x in the expressions

1.  y = sec x    y = 1/ cos pi/3 = 2
    y = 1/ cos -pi = 2
2.  y = sin pi/3 = sqrt 3 over 2
      y = sin -pi/3 = -sqrt 3 over 2
3. y = 1/ tan  pi /3 = sqrt 3 / 3 
     y = 1/ tan  pi /3 = -sqrt 3 / 3 
4.  y = csc x = 1/ sin pi/3 = 2/sqrt 3
    y = csc x = 1/ sin -pi/3 = -2/sqrt 3

odd functions are B, C and D