How can you explain that
sin(180-θ)=sinθ
using the unit circle??
Think about a line that goes from the center of the unit circle to it's perimeter and makes some angle $\displaystyle \theta$ with the x axis.
The sine of $\displaystyle \theta$ is the y coordinate of the intersection of that line and the unit circle because it's radius is 1.
Now what does $\displaystyle 180-\theta$ do. It reflects this line about the y axis. The x coordinate of the intersection with the unit circle is multiplied by -1, and the y coordinate remains the same. Thus the sine of $\displaystyle \theta$ remains the same.