Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

There was a question on my math test that asked for the principal value of $\arcsin(\sin(10)$. My friend says it is $10-3 \pi$ but I say it is $3 \pi -10$. However, taking into account the definition of the range of the inverse sin function, couldnt it be either? Since the answer $A$ has to be such that $\frac{\pi}{2}<A<-\frac{\pi}{2}$.

Can someone clarify this?

share|cite|improve this question
It is $3 \pi - 10$ as you say it. – Sasha Sep 24 '11 at 22:24
$\sin(3\pi-10)=-\sin(10-3\pi)$, so a simple calculation of the sine values should show that one of the two proposed answers must be wrong. – Henning Makholm Sep 24 '11 at 22:25

Note that the value $A$ should satisfy $-\frac{\pi}{2}\leq A\leq \frac{\pi}{2}$; so there are two problems with what your wrote at the end:

  • Inequalities should not be strict; but more importantly:
  • There are no numbers that satisfy what you wrote! You are asking for numbers $A$ that are simultaneously greater than $\frac{\pi}{2}$ and smaller than $-\frac{\pi}{2}$. There's no such numbers.

As Henning notes, the answer cannot be both, because sine gives different answers at those two. However, you can verify that numbers that satisfy $-\frac{\pi}{2}\leq A \lt 0$ will have $\sin(A)\lt 0$, and numbers $A$ that satisfy $0\lt A\leq \frac{\pi}{2}$ will have $\sin(A)\gt 0$. Since you are looking for an angle that has the same value as $\sin(10)$, and $\sin(10)\lt 0$, the answer has to satisfy $-\frac{\pi}{2}\leq A\lt 0$.

If you think about the cycles of $\sin x$, you have a full cycle between $0$ and $2\pi$, one between $2\pi$ and $4\pi$; $10$ lies in the latter; and in fact, $3\pi\lt 10 \lt 4\pi$, so $\sin(10)$ will be negative, hence $\arcsin(\sin 10)$ will be negative. If you think about the graph of sine, you should see that the value will be either $3\pi-10$ or $10-4\pi$, whichever lies in the correct interval, and so the answer is 3\pi-10$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.