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Cosine

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Cosine

Study Guide

Key Definition

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. It is often denoted as $\cos(\theta) = \frac{a}{h}$.

Important Notes

The cosine function is periodic with a period of $2\pi$.
The cosine of 0 is 1, and the cosine of $\frac{\pi}{2}$ (or 90°) is 0.
Cosine function is even, which means $\cos(x) = \cos(-x)$.

Mathematical Notation

$\cos(\theta)$ denotes the cosine of an angle.

$a$ (adjacent side) and $h$ (hypotenuse) represent the lengths of the sides.

Remember to use proper notation when solving problems.

Why It Works

Cosine works because it's based on the inherent properties of right-angled triangles and the ratios of their sides. It is a fundamental concept in trigonometry and helps describe circular motion and wave patterns.

Remember

The cosine ratio remains the same regardless of the size of the triangle. So you can always use it to find an angle in a right-angled triangle if you know the lengths of the sides.

Quick Reference

Cosine Ratio:$\cos(\theta) = \frac{a}{h}$

Reciprocal:$\sec(\theta) = \frac{h}{a}$

Understanding Cosine

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In a right-angled triangle, cosine is the ratio of the length of the side adjacent to an angle to the length of the hypotenuse. It is represented as $\cos(\theta) = \frac{a}{h}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If in a right-angled triangle, the length of the adjacent side is 4 and the length of the hypotenuse is 5, what is the cosine of the angle?

Real-World Problem

Question Exercise

Intermediate

Skateboarding Ramp

A skateboarder wants to build a ramp with a hypotenuse of 13 feet and an adjacent side of 5 feet. What will be the angle of the ramp?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find the value of $\cos(0)$ and $\cos(\frac{\pi}{2})$.

Challenge Quiz

Single Choice Quiz

Advanced

Which property of the cosine function is illustrated by the identity $\cos(x) = \cos(-x)$?

Recap

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Review key concepts and takeaways