Properties of Tangent Functions - Algebra-2

1. Fundamental Concepts

Definition: The tangent function, denoted as $$\tan(\theta)$$ , is defined as the ratio of the sine to the cosine of an angle: $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$ .
Periodicity: The tangent function has a period of $$\pi$$ , meaning $$\tan(\theta + \pi) = \tan(\theta)$$ for all $$\theta$$ .
Asymptotes: The tangent function has vertical asymptotes at $$\theta = \frac{\pi}{2} + k\pi$$ where $$k$$ is any integer, because $$\cos(\theta) = 0$$ at these points.

2. Key Concepts

Period: Repeats every $\pi$ radians (period = $\pi$), much shorter than sine/cosine; thus, $\tan(x + \pi) = \tan x$ for all valid $x$.

Amplitude: None (the function has no maximum or minimum value, so amplitude is undefined).

Symmetry: Odd function, satisfying $\tan(-x) = -\tan x$ (symmetric about the origin).

Vertical Asymptotes: Occur at $x = \frac{\pi}{2} + k\pi$ (undefined points), where the graph approaches $+\infty$ from the left and $-\infty$ from the right (or vice versa).

Key Points: Critical points in one period ($-\frac{\pi}{2}, \frac{\pi}{2}$):

$(0, 0)$ (midpoint, crosses the origin)

$(\frac{\pi}{4}, 1)$ (positive value)

$(-\frac{\pi}{4}, -1)$ (negative value)

3. Examples

Easy

Find the values of $\tan(0)$ and $\tan(\frac{\pi}{4})$.

Solution: Use key points/definition:

$\tan(0) = \frac{\sin 0}{\cos 0} = \frac{0}{1} = 0$; $\tan(\frac{\pi}{4}) = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$.

Medium

Determine if $x = \frac{3\pi}{2}$ is in the domain of $y = \tan x$, and find $\tan(-\frac{\pi}{4})$.

Solution:

1. Domain check: $\frac{3\pi}{2} = \frac{\pi}{2} + \pi$ (fits $x = \frac{\pi}{2} + k\pi$ with $k=1$), so $x = \frac{3\pi}{2}$ is not in the domain.

2. Odd function property: $\tan(-\frac{\pi}{4}) = -\tan(\frac{\pi}{4}) = -1$.

Hard

For $y = \tan x$, find all $x$ in $[0, 2\pi]$ where the function is undefined, and solve $\tan x = 0$ in the same interval.

Solution:

1. Undefined points (asymptotes): $x = \frac{\pi}{2} + k\pi$. In $[0, 2\pi]$, $k=0$ gives $x = \frac{\pi}{2}$, $k=1$ gives $x = \frac{3\pi}{2}$.

2. Solve $\tan x = 0$: $\tan x = 0$ when $\sin x = 0$ (and $\cos x \neq 0$). In $[0, 2\pi]$, $x = 0, \pi, 2\pi$.

4. Problem-Solving Techniques

Visual Strategy: Use the unit circle to visualize the values of sine, cosine, and tangent at different angles.
Error-Proofing: Always check the signs of sine and cosine when determining the value of tangent.
Concept Reinforcement: Practice with a variety of angles to understand the periodicity and behavior of the tangent function.