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Tangent Function

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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 3

Remember 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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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

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

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

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

Real-World Problem

Question Exercise

Intermediate

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?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

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

Challenge Quiz

Single Choice Quiz

Advanced

What is the period of the tangent function?

Recap

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Review key concepts and takeaways