Pls Help Explain This Confusing Explaination for sin(a)=sin(180​∘​​−α) for any angle α Can anybody help explain why sin(Z)=sin(θ) in the image that I provided below?
*I put the confusing part in red rectangle. It's clear in the diagram that sin(Z), i.e. ∠XZY, is bigger than sin(θ) and the explanation that they gave is not satisfying enough for me. I get that, for any angle, sin(a)=sin(180​∘​​−α) but still you can clearly see that they are not the same size.

 A: Your red-boxed equation comes from the general identity that is mentioned just above: for any angle $\alpha$, $\sin(\alpha)=\sin (180^\circ -\alpha)$; and because $\angle XZY=180^\circ-\theta$.
The identity $\sin(\alpha)=\sin (180^\circ -\alpha)$ can be derived from the sine of a difference formula:
$\sin(180^\circ-\alpha)=\sin(180^\circ)\cdot\cos(\alpha)-\cos(180^\circ)\sin(\alpha)
=0\cdot\cos(\alpha)-(-1)\cdot\sin(\alpha)=\sin(\alpha)$.
Or you can convince yourself of the identity by noticing that the points on the unit circle corresponding to $\alpha$ and $(180^\circ-\alpha)$ are reflections of each other across the vertical axis.
A: It appears to me that you are confusing the value of the sine of an angle with the value of the angle itself. For a given value of a sine function, there are two values of the corresponding argument: If the function value is in the range $[0,1]$, then the alternate value for the argument $x$ is $\pi-x$; if the function value is in the range $[-1,0]$, then the alternate value for the argument $x$ is $3\pi-x$. To convince yourself of this, view the graph of the function $\sin{x}=y$, where $x\in [0,2\pi]$.
