Trigonometric Functions: The Unit Circle and Their Properties

Chapter 4: Trigonometric Functions

4.2 Trigonometric Functions: The Unit Circle

The unit circle is a fundamental concept in trigonometry, providing a geometric framework for defining the six trigonometric functions. It is a circle with a radius of 1 centered at the origin of the coordinate plane. The position of a point on the unit circle corresponds to an angle measured from the positive x-axis.

The Unit Circle

• The equation of the unit circle is $x^2 + y^2 = 1$.

• Any point $P = (x, y)$ on the unit circle corresponds to an angle $t$ (in radians) measured from the positive x-axis.

• The coordinates $(x, y)$ represent the cosine and sine of the angle $t$, respectively.

The Six Trigonometric Functions

The six trigonometric functions are defined using the coordinates of a point on the unit circle:

• Sine: $\sin t = y$

• Cosine: $\cos t = x$

• Tangent: $\tan t = \frac{y}{x}$, $x \neq 0$

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

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

• Cotangent: $\cot t = \frac{x}{y}$, $y \neq 0$

Example: Finding Values of the Trigonometric Functions

Suppose $P = \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)$ is a point on the unit circle corresponding to angle $t$.

• $\sin t = \frac{1}{2}$

• $\cos t = \frac{\sqrt{3}}{2}$

• $\tan t = \frac{\sin t}{\cos t} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$

• $\csc t = 2$

• $\sec t = \frac{2}{\sqrt{3}}$

• $\cot t = \sqrt{3}$

The Domain and Range of the Sine and Cosine Functions

• Domain: All real numbers, $(-\infty, \infty)$

• Range: $[-1, 1]$

Even and Odd Trigonometric Functions

Trigonometric functions can be classified as even or odd based on their symmetry properties:

• Even functions: $\cos(-t) = \cos t$, $\sec(-t) = \sec t$

• Odd functions: $\sin(-t) = -\sin t$, $\csc(-t) = -\csc t$, $\tan(-t) = -\tan t$, $\cot(-t) = -\cot t$

Fundamental Trigonometric Identities

Several identities are fundamental to trigonometry and are used to simplify expressions and solve equations:

• Pythagorean Identities:

• $\sin^2 t + \cos^2 t = 1$

• $1 + \tan^2 t = \sec^2 t$

• $1 + \cot^2 t = \csc^2 t$

Definition of a Periodic Function

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

Periodic Properties of the Sine and Cosine Functions

• $\sin(t + 2\pi) = \sin t$

• $\cos(t + 2\pi) = \cos t$

• The period of sine and cosine is $2\pi$.

Periodic Properties of the Tangent and Cotangent Functions

• $\tan(t + \pi) = \tan t$

• $\cot(t + \pi) = \cot t$

• The period of tangent and cotangent is $\pi$.

Repetitive Behavior of the Sine, Cosine, and Tangent Functions

The graphs of sine, cosine, and tangent functions repeat their values at regular intervals, reflecting their periodic nature.

Using a Calculator to Evaluate Trigonometric Functions

• Set the calculator to the correct mode: degrees or radians (for the unit circle, use radians).

• Use the SIN, COS, and TAN keys to evaluate trigonometric functions.

• Consult your calculator's manual for specific instructions.

Example: $\sin \frac{\pi}{4} \approx 0.7071$

Example: $\sec 0.1 \approx 1.0025$