So I memorized all the trig functions in high school. Now I'm applying them and wanted to know why $\sin\theta = opp/hyp$?
I understand on the unit circle, when I trace the y values I see that $y = \sin\theta$. But when the hypotenuse is other than 1 why does $y = hyp\cdot\sin\theta$?
1 Answer
$sin\theta$ is defined as $opp/hyp$. In a unit circle $hyp=1$, so we get $y=sin\theta$, where $y$ is the opposite side. In other triangles the value of $hyp$ is other than $1$. However as the angles in the triangle inscribed in the unit circle is same as that of the new triangle, we conclude that these two triangles are similiar.
So, their sides are proportional, i.e, $opp/hyp$ of first triangle equals $opp/hyp$ of the second triangle. Since $opp/hyp$ of first traingle=$sin\theta$, $opp/hyp$ of the second triangle is also equal to $sin\theta$. We can now easily see that $opp=hyp sin\theta$.
