Circular functions: Graphs, Composites, and Transformations
Sine and Cosine
Sine and cosine are very similar, and share properties:
- Both have a period of $2\pi$ radians or 360°
- Their ranges are limited to [-1, 1]
- They are continuous and differentiable everywhere
The graphs of $y = \sin x$ and $y = \cos x$ are as follows:
The cosine function is essentially a horizontal shift of the sine function by $\frac{\pi}{2}$ radians or 90°.
Tangent
The tangent function is defined as $\tan x = \frac{\sin x}{\cos x}$. Its properties include:
- A period of $\pi$ radians or 180°
- An undefined value at odd multiples of $\frac{\pi}{2}$ (i.e. $\frac\pi2, \frac{3\pi}2, \frac{5\pi}2, $ etc.)
- An unbounded range (all real numbers)
The graph of $y = \tan x$ looks like this:
Students often forget that the tangent function has vertical asymptotes at $x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}$, etc.
Transformations of Trigonometric Functions
Trigonometric functions can be transformed like any other function.
Vertical and Horizontal Stretches
- Vertical stretch: $y = af(x)$ where $|a| > 1$
