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Graphing Cosine Function

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Graphing Cosine Function

Study Guide

Key Definition

The cosine function $y = \cos(x)$ is a periodic function with a period of $2\pi$ and ranges from $-1$ to $1$.

Important Notes

The period of $y = \cos(x)$ is $2\pi$.
The domain of $y = \cos(x)$ is $(-\infty, \infty)$.
The range of $y = \cos(x)$ is $[-1, 1]$.
The function is even and symmetric about the y-axis.
Maxima occur at $x = 2n\pi$ and minima at $x = \pi + 2n\pi$ where $n$ is an integer.

Mathematical Notation

$\cos(x)$ is the cosine function

$\pi$ is approximately 3.14159

$2\pi$ is the period of the cosine function

Remember to use proper notation when solving problems

Why It Works

The cosine function is derived from the unit circle, where $\cos(\theta)$ represents the x-coordinate of a point on the circle.

Remember

The cosine function is even, meaning $\cos(-x) = \cos(x)$.

Quick Reference

Domain:$(-\infty, \infty)$

Range:$[-1, 1]$

Period:$2\pi$

Amplitude:$1$

Understanding Graphing Cosine Function

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $y = \cos(x)$ showing basic wave shape.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the amplitude of $y = \cos(x)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You're designing a Ferris wheel where the height follows a cosine function $h = 10 \cos(\frac{\pi}{15}t) + 15$. What is the maximum height?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

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

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following represents a phase shift of $\frac{\pi}{4}$ to the right?

Recap

Watch & Learn

Review key concepts and takeaways