Domain and Range of Trigonometric Functions
What Are Domain and Range?
Before diving into trigonometric functions, let's recall two fundamental concepts:
- Domain: The set of all possible input values (x-values) for which a function is defined.
- Range: The set of all possible output values (y-values) that a function can produce.
For trigonometric functions, the domain tells us which angles we can plug in, and the range tells us what values come out.
The Unit Circle Connection
Remember that on the unit circle (a circle with radius 1 centered at the origin), for any angle $\theta$:
$\cos\theta = x\text{-coordinate}$ $\sin\theta = y\text{-coordinate}$
$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
Since the unit circle has radius 1, the coordinates always satisfy $-1 \leq x \leq 1$ and $-1 \leq y \leq 1$. This is the geometric reason why sine and cosine have bounded ranges!