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?
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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. |
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The usual convention is that $\sin^2(X)=(\sin(X))^2$. So for your example $1/4$ is correct |
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$\sin^2(30)=(\sin(30))^2$ so it is equal $(1/2)^2=1/4$ |
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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: $$\frac12 (1 - \cos (2x)) = \frac12 (1 - (1 - 2 \sin^2 x)) = \frac12 (2 \sin^2 x) = \sin^2 x.$$ |
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