Graphing Tangent Function - Varsity Tutors

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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 3.14159Remember 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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Beginner Explanation

The tangent function $\tan(x)$ is a trigonometric function with a period of $\pi$.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

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

A$\pi$B$2\pi$C$\frac{\pi}{2}$D$\frac{2\pi}{3}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

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.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Quick Quiz

Single Choice QuizBeginner

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

A$1$B$0$C$\sqrt{2}$D$\frac{1}{\sqrt{2}}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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