Tangent (of An Angle) - Varsity Tutors

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Tangent (of an angle)

Study Guide

Key Definition

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

Important Notes

  • Tangent is abbreviated as $\tan$
  • The reciprocal of tangent is cotangent: $\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}$
  • The tangent ratio is constant for a given angle regardless of the triangle size
  • Angles can be measured in degrees or radians
  • Tangent can be thought of as a function that varies with the angle measure

Mathematical Notation

$\tan(\theta)$ represents the tangent of angle $\theta$$\cot(\theta)$ represents the cotangent of angle $\theta$$\frac{a}{b}$ is a fractionRemember to use proper notation when solving problems

Why It Works

The tangent function relates the angle of a right triangle to the ratio of its sides, allowing for the calculation of unknown side lengths or angles.

Remember

Tangent is useful for finding the slope of an angle when considering the rise over run in a triangle.

Quick Reference

Tangent Formula:$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$Cotangent Formula:$\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}$

Understanding Tangent (of an angle)

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Beginner Explanation

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

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

What is $\tan(\theta)$ in a triangle where the opposite side is 6 and the adjacent side is 8?

A$\frac{3}{4}$B$\frac{4}{3}$C$\frac{6}{8}$D$\frac{8}{6}$Check 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 at a park and need to find the height of a tree. You stand 10 meters away from the base of the tree and measure the angle of elevation to the top as $30^\\circ$. Calculate the height of the tree using tangent.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

In a right triangle, if the tangent of an angle is $\frac{3}{4}$, what are the possible lengths of the opposite and adjacent sides?

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

Challenge Quiz

Single Choice QuizAdvanced

Find $\tan(\theta)$ if the opposite side is $\sqrt{3}$ and the adjacent side is 1.

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

Recap

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