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Graphing Tangent Function

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Graphing Tangent Function

Study Guide

Key Definition

The tangent function is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and is used to graph angles in trigonometry.

Important Notes

The tangent function is undefined when $\cos(x) = 0$
The graph has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$
The period of the tangent function is $\pi$
The tangent function is an odd function, meaning it has origin symmetry
Zeroes of the tangent function occur at $n\pi$

Mathematical Notation

$\tan(x)$ is the tangent function

$\sin(x)$ is the sine function

$\cos(x)$ is the cosine function

$\frac{a}{b}$ represents a fraction

$\pi$ represents the constant pi, approximately 3.14159

Remember to use proper notation when solving problems

Why It Works

The tangent function's graph is based on its definition $\tan(x) = \frac{\sin(x)}{\cos(x)}$, which inherently includes properties like periodicity and asymptotes due to the behavior of sine and cosine.

Remember

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

Quick Reference

Domain:$x \in \mathbb{R}, x \neq \frac{\pi}{2} + n\pi$

Range:$(-\infty, \infty)$

Period:$\pi$

Symmetry:Origin (odd function)

Understanding Graphing Tangent Function

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The tangent function $\tan(x)$ is a trigonometric function with a period of $\pi$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the period of the tangent function $\tan(x)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp is designed with an angle $\theta$ such that $\tan(\theta) = \frac{3}{4}$. Calculate the height of the ramp if the base is 8 feet.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the x-intercepts of $y = \tan(x)$ on the interval $[0, 2\pi]$.

Quick Quiz

Single Choice Quiz

Beginner

What is the value of $\tan(x)$ at $x = \frac{\pi}{4}$?

Recap

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Review key concepts and takeaways