Here is a link to the Wikipedia site which has a chart of all the relationships between theta and the different trig functions:
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
My question is, how exactly is that when $\theta = \arcsin$ then the opposite side is equal to $x$ but when $\theta = \arccos$ then the opposite side equals $\sqrt{1-x^{2}}$. How are the sides determined? Could you provide a picture describing how the sides change with $\theta$?
Best Answer
If you look at different columns, you should be able to guess. Here it is how it's done: For $\theta=\arcsin(x)$ we take $\sin$ of both sides, so $\sin(\theta)=\sin(\arcsin(x))=x$. In a right angle triangle, the sine of an angle is the opposite side divided by the hypotenuse. If the hypotenuse is 1, the sine of the angle is equal to the opposite side. Since $\sin(\theta)=x$, it means that the opposite side is $x$. Similarly for cosine function, the cosine of the angle is defined as the neighboring side divided by the hypotenuse. If $\theta=\arccos(x)$ then $\cos(\theta)=\cos(\arccos(x))=x$, so it means that in this case $x$ is the length of the adjacent side in a right angle triangle where the hypotenuse length is 1. Applying Pythagoras's theorem, the opposite side is $\sqrt{1-x^2}$