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

Let's take the sine of $30^\circ$ which is one-half. If you take $\sin^2(30^\circ)$, would that just be the sine of $900$? Or would it be equal to one-quarter, or would it be equal to something completely different?

share|cite|improve this question
$\sin^2 x =(\sin x)^2$ – pedja Mar 22 '12 at 20:40
Understanding what is sin(x) may make things clearer. One source is: – NoChance Mar 22 '12 at 22:45
up vote 5 down vote accepted

I think you are confused with the following notation:

$$\sin ^2(x)=\sin x \cdot \sin x \neq \sin(x^2) ~~\mbox{very often.}$$

So, $\sin^2(30^\circ)=\dfrac 1 4$.

And, $\sin 900^\circ$ is not untractable either.

$$\sin 900^\circ=\sin 5 \pi=0$$

I am being nitpicky now:

When you write $900$, I assume that you mean $900^\circ$. But, in Mathematics, it is a convention that $900$ means $900^c=900$ radians. For definition of a radian and other details, you may want to look up Wikipedia.

share|cite|improve this answer
Actually, there are infinitely many solutions to $\sin^2(x) = \sin(x^2)$, as should be obvious if you plot both sides as functions of $x$. The smallest positive solution appears to be about 1.3644129631252872330 (in radians). – Ilmari Karonen Mar 22 '12 at 20:53
Thank you. I was not being careful.... – user21436 Mar 22 '12 at 20:55
@IlmariKaronen I have now edited to make it unambiguous. – user21436 Mar 22 '12 at 20:56

The usual convention is that $\sin^2(X)=(\sin(X))^2$. So for your example $1/4$ is correct

share|cite|improve this answer

As many people have pointed out by now, $\sin^2 x$ is simply a "nickname" for $(\sin x)^2$. Therefore, $\sin^2\ 30 = (\sin 30)^2 = (1/2)^2 = 1/4$.

As it happens, though, there is another useful thing we can say about $\sin^2 x$:

$$\sin^2 x = (\sin x)^2 = \frac12 (1 - \cos (2 x)).$$

We can see this using the double-angle formula for cosines:

$$\frac12 (1 - \cos (2x)) = \frac12 (1 - (1 - 2 \sin^2 x)) = \frac12 (2 \sin^2 x) = \sin^2 x.$$

share|cite|improve this answer
+1, though I would have put $\sin^2(x)=1-\cos^2(x)$ before $\frac12 (1 - \cos (2 x))$ – Henry Mar 23 '12 at 0:46

$\sin^2(30)=(\sin(30))^2$ so it is equal $(1/2)^2=1/4$

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.