Question

why inverse(sin(-x)) equivalent with -(inverse(sin(x)))

Answer 1

because both the inside function and the outside one are odd

Answer 2

\[\sin^{-1} (-x)=-\sin^{-1} x\]
because \(\sin^{-1} x\) is an odd function

Answer 3

please tell me, what do you mean they both odd function?

Answer 4

\[\sin(-x)=-\sin(x)\] and
\[\sin^{-1}(-x)=-\sin{-1}(x)\]
therefore
\[\sin^{-1}(\sin(-x))=\sin^{-1}(-sin(x))=-\sin^{-1}(\sin(x))\]

Answer 5

if
f(-x)=-f(x) then the function is odd

Answer 6

definition :

\[f \text { is odd if } \f(-x)=-f(x)\]

Answer 7

thanks , tan is odd function to right?

Answer 8

yeah tan x is odd

Answer 9

sorry, what about cos?

Answer 10

cos x is even
\[cos (x)=cos(-x)\]

Answer 11

oh I see what do you mean. interesting..

Answer 12

what about cos\[\sec ^{-1}(-x)=\pi+\sec^(-1)\]

Answer 13

it may help you give you an idea what that mean graphically with some examples
Even functions:
y=x^2
http://www.wolframalpha.com/input/?i=plot%20y%3Dx%5E2&t=crmtb01
y=x^2-x^4
http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E2-x%5E4
y=cos x
http://www.wolframalpha.com/input/?i=plot+y%3Dcos+x
notice they are all symmetric about x=0. this is a direct result of the fact that \[f(-x)=f(x)\]

Answer 14

\[\sec^−1(−x)=π+\sec ^{-1}\]