Part 3/5:
When considering an angle shifted by (2\pi), we find that the new angle drawn from the same initial line returns to the same terminal side. Thus, the ratios (or sine and cosine values) remain consistent, confirming the established identities.
Inferring Half-Angle Identities
The tangent angle identity involving halved angles, particularly ( \frac{\pi}{2} ) (which is equal to 90 degrees), is foundational. Here, the sine and cosine of ( \frac{\pi}{2} ) minus another angle, ( \theta ), can be expressed using the complementary angle relationships:
[
\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)
]
[
\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)
]