Graphs and Properties of Tangent and Cotangent Functions

Trigonometric Functions

Tangent and Cotangent Functions: Properties and Graphs

The tangent and cotangent functions are fundamental trigonometric functions with unique properties and graphs. Understanding their shapes, asymptotes, and zeroes is essential for analyzing periodic phenomena and solving equations involving angles.

• Definition: The tangent function is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$, and the cotangent function is $\cot(x) = \frac{\cos(x)}{\sin(x)}$.

• Domain: $\tan(x)$ is undefined where $\cos(x) = 0$ (i.e., $x = \frac{\pi}{2} + k\pi$ for integer $k$). $\cot(x)$ is undefined where $\sin(x) = 0$ (i.e., $x = k\pi$ for integer $k$).

• Periodicity: Both functions have a period of $\pi$.

• Asymptotes: Vertical asymptotes occur at the points where the function is undefined.

• Zeroes: $\tan(x)$ has zeroes at $x = k\pi$; $\cot(x)$ has zeroes at $x = \frac{\pi}{2} + k\pi$.

Example: The graph of $y = \tan(x)$ has vertical asymptotes at $x = \frac{\pi}{2}, \frac{3\pi}{2}, ...$ and passes through the origin.

Example: The graph of $y = \cot(x)$ has vertical asymptotes at $x = 0, \pi, 2\pi, ...$ and passes through $x = \frac{\pi}{2}, \frac{3\pi}{2}, ...$.

Graphing Tangent and Cotangent Functions Using Transformations

Transformations such as vertical and horizontal shifts, stretches, and compressions can be applied to tangent and cotangent functions. The key to graphing these functions is to track five characteristic points: zeroes, asymptotes, and points at $\pm\frac{\pi}{4}$ from the zero.

• General Form: $y = a \tan(bx - c) + d$ and $y = a \cot(bx - c) + d$

• Steps:

  1. Identify the period: $\frac{\pi}{|b|}$

  2. Locate vertical asymptotes

  3. Mark zeroes

  4. Plot points at $\pm\frac{1}{4}$ period from zero

  5. Apply vertical and horizontal shifts

Example: Graph $y = \tan(x)$ and $y = \tan(x - \frac{\pi}{4})$ to observe the horizontal shift.

Equation of a Line Using Tangent

Given the angle a line makes with the x-axis (base) and a point through which it passes, the equation of the line can be determined using the tangent function. The slope of the line is given by $m = \tan(\theta)$, where $\theta$ is the angle with the x-axis.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

• Application: If a line passes through $(4, 5)$ and makes a $60^\circ$ angle with the x-axis, its slope is $m = \tan(60^\circ) = \sqrt{3}$.

Example: Find the equation of a line passing through $(4, 5)$ with an angle of $60^\circ$ to the x-axis.

• Slope: $m = \tan(60^\circ) = \sqrt{3}$

• Equation: $y - 5 = \sqrt{3}(x - 4)$

Summary Table: Properties of Tangent and Cotangent Functions

Function	Zeroes	Asymptotes	Period
$\tan(x)$	$x = k\pi$	$x = \frac{\pi}{2} + k\pi$	$\pi$
$\cot(x)$	$x = \frac{\pi}{2} + k\pi$	$x = k\pi$	$\pi$

Key Formulas

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$

• $\cot(x) = \frac{\cos(x)}{\sin(x)}$

• Slope of a line: $m = \tan(\theta)$

Additional info: The checkpoint and examples referenced are standard in precalculus trigonometry, focusing on graphing and applications of tangent and cotangent functions.