Question

examine the angle and the point where the terminal side intersects the circle. what is the sine of the angle?

Answer 1

In order to find sin(theta), we need to find the sine of the red angle:

where the hypotenuse is given by

$\begin{gathered} c=\sqrt[]{(\frac{\sqrt[]{3}}{2}})^2+(\frac{1}{2})^2 \\ c=\sqrt[]{\frac{3}{4}+\frac{1}{4}} \\ c=\sqrt[]{1} \\ c=1 \end{gathered}$

then, sine of this angle is equal to

$\sin \varphi=\frac{\frac{1}{2}}{1}=\frac{1}{2}$

On the other hand,

$\sin \theta=-\sin \varphi$

then, the answer is

$\sin \theta=-\frac{1}{2}$

which corresponds to option 3.