Unit Circle Approach and Properties of Trigonometric Functions

Trigonometric Functions: Unit Circle Approach

Definition and Exact Values Using the Unit Circle

The unit circle approach is fundamental in precalculus for defining and evaluating trigonometric functions. The unit circle is a circle of radius 1 centered at the origin in the coordinate plane. Each real number t corresponds to a point P = (a, b) on the unit circle, and the trigonometric functions are defined based on the coordinates of this point.

• Sine function: $\sin t = b$

• Cosine function: $\cos t = a$

• Tangent function: $\tan t = \frac{b}{a}$, $a \neq 0$

• Cosecant function: $\csc t = \frac{1}{b}$, $b \neq 0$

• Secant function: $\sec t = \frac{1}{a}$, $a \neq 0$

• Cotangent function: $\cot t = \frac{a}{b}$, $b \neq 0$

Example: Finding Exact Values

Given P = (–3/2, –1/2) on the unit circle:

• $\sin t = -\frac{1}{2}$

• $\cos t = -\frac{3}{2}$

• $\tan t = \frac{-1/2}{-3/2} = \frac{1}{3}$

• $\csc t = \frac{1}{-1/2} = -2$

• $\sec t = \frac{1}{-3/2} = -\frac{2}{3}$

• $\cot t = \frac{-3/2}{-1/2} = 3$

Trigonometric Functions for Angles on Circles of Radius r

For an angle θ in standard position, let P = (a, b) be any point on the terminal side of θ that is also on the circle $x^2 + y^2 = r^2$. The trigonometric functions are defined as:

• $\sin \theta = \frac{b}{r}$

• $\cos \theta = \frac{a}{r}$

• $\tan \theta = \frac{b}{a}$, $a \neq 0$

• $\csc \theta = \frac{r}{b}$, $b \neq 0$

• $\sec \theta = \frac{r}{a}$, $a \neq 0$

• $\cot \theta = \frac{a}{b}$, $b \neq 0$

Special Right Triangles and Trigonometric Values

Special right triangles, such as the 45-45-90 and 30-60-90 triangles, are used to find exact values of trigonometric functions for common angles.

• 45-45-90 Triangle: Both legs are equal, hypotenuse is $\sqrt{2}$ times a leg.

• 30-60-90 Triangle: Shorter leg is half the hypotenuse, longer leg is $\sqrt{3}$ times the shorter leg.

Domain and Range of Trigonometric Functions

Domain of Trigonometric Functions

The domain of a function is the set of all possible input values (θ) for which the function is defined.

• Sine and Cosine: All real numbers

• Tangent and Secant: All real numbers except odd integer multiples of $\frac{\pi}{2}$ (90°)

• Cotangent and Cosecant: All real numbers except integer multiples of $\pi$ (180°)

Range of Trigonometric Functions

The range of a function is the set of all possible output values.

• Sine and Cosine: $-1 \leq \sin \theta \leq 1$, $-1 \leq \cos \theta \leq 1$

• Secant and Cosecant: $\sec \theta \leq -1$ or $\sec \theta \geq 1$, $\csc \theta \leq -1$ or $\csc \theta \geq 1$

• Tangent and Cotangent: $-\infty < \tan \theta < \infty$, $-\infty < \cot \theta < \infty$

Summary Table: Domains and Ranges

The following table summarizes the domains and ranges of the six trigonometric functions:

Function	Symbol	Domain	Range
Sine	$f(\theta) = \sin \theta$	All real numbers	All real numbers from –1 to 1, inclusive
Cosine	$f(\theta) = \cos \theta$	All real numbers	All real numbers from –1 to 1, inclusive
Tangent	$f(\theta) = \tan \theta$	All real numbers, except odd integer multiples of $\frac{\pi}{2}$ (90°)	All real numbers
Cosecant	$f(\theta) = \csc \theta$	All real numbers, except integer multiples of $\pi$ (180°)	All real numbers greater than or equal to 1 or less than or equal to –1
Secant	$f(\theta) = \sec \theta$	All real numbers, except odd integer multiples of $\frac{\pi}{2}$ (90°)	All real numbers greater than or equal to 1 or less than or equal to –1
Cotangent	$f(\theta) = \cot \theta$	All real numbers, except integer multiples of $\pi$ (180°)	All real numbers

Periodic Properties of Trigonometric Functions

Definition of Periodic Functions

A function f is periodic if there is a positive number p such that $f(\theta + p) = f(\theta)$ for all θ in the domain. The smallest such p is called the fundamental period of f.

Periodic Properties of Trigonometric Functions

• $\sin(\theta + 2\pi) = \sin \theta$

• $\cos(\theta + 2\pi) = \cos \theta$

• $\tan(\theta + \pi) = \tan \theta$

• $\csc(\theta + 2\pi) = \csc \theta$

• $\sec(\theta + 2\pi) = \sec \theta$

• $\cot(\theta + \pi) = \cot \theta$

Even-Odd Properties of Trigonometric Functions

Definition and Properties

Trigonometric functions exhibit even-odd properties, which describe their symmetry with respect to the y-axis or origin.

• $\sin(-\theta) = -\sin \theta$ (odd)

• $\cos(-\theta) = \cos \theta$ (even)

• $\tan(-\theta) = -\tan \theta$ (odd)

• $\csc(-\theta) = -\csc \theta$ (odd)

• $\sec(-\theta) = \sec \theta$ (even)

• $\cot(-\theta) = -\cot \theta$ (odd)

Example: Using Even-Odd Properties

• $\sin(-30^\circ) = -\sin(30^\circ)$

• $\cos(-\frac{3\pi}{2}) = \cos(\frac{3\pi}{2})$

• $\cot(-\frac{5\pi}{4}) = -\cot(\frac{5\pi}{4})$

• $\tan(-\frac{29\pi}{4}) = -\tan(\frac{29\pi}{4})$

Summary

The unit circle approach provides a geometric foundation for defining and evaluating trigonometric functions. Understanding the domains, ranges, periodicity, and symmetry properties of these functions is essential for solving problems in precalculus and beyond.