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 $|a| > 1$
- Vertical compression: $y = af(x)$ where $0 < |a| < 1$
- Horizontal stretch: $y = f(bx)$ where $0 < |b| < 1$
- Horizontal compression: $y = f(bx)$ where $|b| > 1$
Remember that horizontal transformations work in the "opposite" way you might expect. A larger $b$ value leads to a compression, not a stretch!
Reflections
- Reflection over x-axis: $y = -f(x)$
- Reflection over y-axis: $y = f(-x)$
Translations
- Horizontal translation: $y = f(x - h)$ (right $h$ units if $h > 0$)
- Vertical translation: $y = f(x) + k$ (up $k$ units if $k > 0$)
Graphing transformations
Applied to a function like $\sin x$, we get:
$f(x) = a \sin(b(x+c)) + d$
Where:
- $a$ affects the amplitude
- $b$ affects the period
- $c$ affects the phase shift
- $d$ affects the vertical shift
This works exactly the same for cosine, just be aware $\cos x$ starts at $(1, 0)$ instead of $(0, 0)$.
Let's consider the function $f(x) = 2\sin(3(x-\frac{\pi}{6})) + 1$
Here:
- $a = 2$ (amplitude is 2)
- $b = 3$ (period is $\frac{2\pi}{3}$)
- $c = -\frac{\pi}{6}$ (phase shift is $\frac{\pi}{6}$ to the right)
- $d = 1$ (vertical shift of 1 unit up)
Compared with the original $\sin x$ graph, this graph looks like:
