Part 2/5:
These properties illustrate that when angles are extended beyond their standard range, the sine and cosine values repeat, leading to their cyclical nature.
Proving the Sine and Cosine Identities
To prove these identities, visualize the Cartesian plane with X and Y axes. By drawing an angle ( \theta ) from the origin, the adjacent and opposite sides of the right triangle can be identified. The cosine of ( \theta ), given by the ratio of the adjacent side to the hypotenuse, can be stated as:
[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
]
On the other hand, the sine can be defined as:
[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
]