Part 2/5:
These identities can be proven visually using the unit circle or a graph. The fundamental idea is that the sine function is odd while the cosine function is even.
Visualizing the Graphs
To intuitively prove these identities, one can refer to graphs of the sine and cosine functions. Consider the sine function, which represents the y-coordinate of points on the unit circle. If we take a positive angle θ, the corresponding point on the unit circle has a y-coordinate that reflects in the negative y-axis when considering a negative angle -θ.
For example, if θ is located at (x, y) on the unit circle, then -θ will correspond to the point (x, -y). From this relation, it’s clear that:
sin(-θ) = -y = -sin(θ)