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Amplitude and Period of Sine and Cosine Functions

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Amplitude and Period of Sine and Cosine Functions

Study Guide

Key Definition

The amplitude of $y = a \sin ( x )$ and $y = a \cos ( x )$ represents half the distance between the maximum and minimum values of the function. Amplitude = $| a |$. The period of $y = a \sin ( b x )$ and $y = a \cos ( b x )$ is given by Period = $2 \pi / | b |$.

Important Notes

The amplitude and period are fundamental properties of sine and cosine functions.
Amplitude measures the 'height' of the function and period measures the 'length' of one complete cycle.

Mathematical Notation

$| a |$ denotes the absolute value of a.

$\sin$ stands for sine function.

$\cos$ stands for cosine function.

$2 \pi / | b |$ calculates the period of the function.

Remember to use proper notation when solving problems

Why It Works

The amplitude and period of sine and cosine functions help us understand the 'shape' of these functions and how they behave over time or across the x-axis. The amplitude tells us how 'tall' or 'short' the function is, while the period tells us how 'long' one complete cycle of the function is.

Remember

The amplitude is always a positive value, while the period is determined by the absolute value of the coefficient of x in the sine or cosine function.

Quick Reference

Amplitude:$| a |$

Period:$2 \pi / | b |$

Understanding Amplitude and Period of Sine and Cosine Functions

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Video explanation of this concept

Beginner

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Beginner Explanation

The amplitude and period are two fundamental properties of sine and cosine functions that help us understand their behavior.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the amplitude and period of the function $y = 5 \cos ( \frac{x}{4} )$?

Real-World Problem

Question Exercise

Intermediate

Ferris Wheel Scenario

A ferris wheel completes one full rotation every 10 minutes. The height of a passenger car from the ground is modeled by the function $y = 10 \sin ( \frac{\pi x}{5} )$, where x is the time in minutes and y is the height in meters. What is the amplitude and period of this function?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given the function $y = 3 \sin ( 2 \pi x )$, can you determine its amplitude and period?

Challenge Quiz

Single Choice Quiz

Advanced

What is the amplitude and period of the function $y = -4 \cos ( \frac{\pi x}{2} )$?

Recap

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