Relationships Between Trigonometric Functions
Symmetries of Trigonometric Functions
Sine Function Symmetry
The sine function has symmetry:
$$\sin(\pi - \theta) = \sin(\theta)$$
This relationship is derived from the fact that the sine function is even about the line $x = \frac \pi2 $ graphically.
This means that
$$\sin(\frac{\pi}{2}-u)=\sin(\frac{\pi}{2}+u)$$
For all $u \in \mathbb{R}$
Using the substitution $ = x - \frac{\pi}{2}$:
$$\sin(\frac{\pi}{2} - (x - \frac{\pi}{2}))=\sin(\frac{\pi}{2}+ x - \frac{\pi}{2})$$
$$\sin(\pi - x)=\sin(x)$$
ExampleFor instance, $\sin(60°) = \sin(120°)$ because $120° = 180° - 60° = \pi - 60°$.
NoteThis symmetry property of sine is related to its odd function nature, where $\sin(-\theta) = -\sin(\theta)$.
Cosine Function Symmetry
The cosine function has a similar symmetry property:
$$\cos(\pi - \theta) = -\cos(\theta)$$
This relationship is derived from the fact that the cosine function is odd about the line $x = \frac{\pi}{\2}$ in its graph.
Therefore, we can say that for all $u\in \mathbb{R}$:
$$\cos(\frac{\pi}{2}-u)=-\cos(\frac{\pi}{2}+u)$$
Therefore, using the substitution $u = x-\frac{\pi}{2}$
$$\cos(\frac{\pi}{2}-(x-\frac{\pi}{2}))=-\cos(\frac{\pi}{2}+(x-\frac{\pi}{2}))$$
$$\cos(\pi - x)=-\cos(x)$$
hence the identity for symmetry is proven
ExampleFor example, $\cos(60°) = -\cos(120°)$, as $120° = 180° - 60° = \pi - 60°$.
NoteThe cosine function is an even function, meaning $\cos(-\theta) = \cos(\theta)$, which is related to but distinct from this symmetry property.
Tangent Function Symmetry
The tangent function exhibits the following symmetry:
$$\tan(\pi - \theta) = -\tan(\theta)$$
