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Tangent Function

Study Guide

Key Definition

The tangent function is defined as the ratio of the sine and cosine of an angle: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Important Notes

  • The tangent function is periodic with a period of $180^\circ$ ($\pi$ radians)
  • Tangent is undefined when $\cos(\theta) = 0$
  • In Quadrant II, tangent is negative since $\sin(\theta)$ is positive and $\cos(\theta)$ is negative
  • In Quadrant III, tangent is positive since both $\sin(\theta)$ and $\cos(\theta)$ are negative
  • In Quadrant IV, tangent is negative since $\sin(\theta)$ is negative and $\cos(\theta)$ is positive

Mathematical Notation

$\tan(\theta)$ is the tangent function$\sin(\theta)$ is the sine function$\cos(\theta)$ is the cosine function$\frac{1}{2}$ represents a fraction$\sqrt{3}$ represents the square root of 3Remember to use proper notation when solving problems

Why It Works

The tangent function works by relating the angles in a circle to the ratios of the sides in right triangles using $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Remember

For key angles, remember the tangent values: $\tan(0^\circ) = 0$, $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, $\tan(45^\circ) = 1$, $\tan(60^\circ) = \sqrt{3}$, $\tan(90^\circ)$ is undefined.

Quick Reference

Tangent of 45°:$1$Tangent of 60°:$\sqrt{3}$Tangent of 0°:$0$Tangent of 90°:Undefined

Understanding Tangent Function

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concept. Use space or enter to play video.concept thumbnailBeginner

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Beginner Explanation

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

What is $\tan(45^\circ)$?

A$1$B$\sqrt{3}$C$0$DUndefinedCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

Imagine you are measuring the height of a tree using a stick. If the angle of elevation to the top of the tree is $30^\circ$ and the distance from the stick to the tree is 10 meters, what is the height of the tree?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

If $\tan(\theta) = \frac{3}{4}$ and $\cos(\theta) > 0$, what is $\sin(\theta)$?

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

Challenge Quiz

Single Choice QuizAdvanced

What is the period of the tangent function?

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

Recap

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