Lesson 4.2

The Cosine Wave

The cosine wave is the sine wave's twin — identical in shape, just shifted along the -axis. Understanding how they relate unlocks the entire family of sinusoidal transformations.

Introduction

Just as sine tracks the -coordinate around the unit circle, cosine tracks the -coordinate. The resulting graph has the exact same shape — but it starts at its maximum instead of at zero.

Past Knowledge

You graphed and identified its five key points per period.

Today's Goal

Graph and understand it as a horizontally shifted sine wave.

Future Success

All sinusoidal transformations (amplitude, period, shifts) apply identically to both sine and cosine.

Key Concepts

The Graph of

Instead of starting at like sine, cosine starts at its maximum because — the -coordinate of the starting point on the unit circle.

Purple = cos(x) · Gray dashed = sin(x)

Cosine's Quarter-Period Pattern

The cosine wave follows: max → zero → min → zero → max.



Pattern	max	zero	min	zero	max

The Phase Relationship

Cosine is sine shifted left by :

This means cosine “leads” sine by a quarter period. Every technique you learn for sine applies equally to cosine.

Worked Examples

Basic

Key Points of Cosine

Question: What is ?

Step 1: is the halfway point of one period (the second key point after the zero).

Step 2: The cosine pattern at the halfway point is min, so .

Final Answer:

Intermediate

Comparing Sine and Cosine

Question: At what -values in do and have the same value?

Step 1: Set , which means .

Step 2: at and .

Final Answer: and

Advanced

Rewriting Cosine as Sine

Question: Express as a sine function.

Step 1: Cosine “leads” sine by . This means the cosine graph is the sine graph shifted left by .

Step 2: A leftward shift adds to the input: