The Graph of Cosine
Simple Explanation
y = cos x is the same shape of periodic wave as sine, but shifted: it starts at its maximum value (1) when x=0, rather than starting at the origin.
Why Do We Need It?
Cosine and sine together describe every possible phase of periodic motion β cosine is simply sine "started a quarter-cycle early."
See It
A smooth wave oscillating between -1 and 1, starting at its maximum value of 1 when x=0, repeating every 2 pi
Formula
Properties of y = cos x
Domain: all real numbers. Range: [β1, 1]. Period: 2Ο
y=cosx oscillates smoothly between β1 and 1, repeating every 2Ο, starting at its maximum value (1) when x=0.
- x
- β the input angle, in radians
When to use it: Whenever identifying or sketching the basic cosine graph and its key features.
Worked Example
Evaluate cosine at its maximum and minimum
Find cos(0) and cos(Ο), and describe how the graph behaves between these two x-values.
Why Does This Work?
At x=0, the point on the unit circle is exactly (1,0) β its x-coordinate (cosine) is at its largest possible value, 1. As x increases toward Ο, the point sweeps around to (β1,0), where the x-coordinate reaches its smallest possible value, β1.
Real-Life Example
Modeling a pendulum released from its highest point
A pendulum is pulled to one side and released from rest, then swings back and forth.
Since it starts at maximum displacement (not at the center), its motion over time is modeled by a cosine curve rather than a sine curve.
Practice
Find cos(Ο/3).
EasyCommon mistake
Confusing the starting value of sine (0, at the origin) with cosine (1, at its maximum) when x=0.
Quick Review
- y=cosx: domain all reals, range [β1,1], period 2Ο.
- Starts at its maximum value (1) when x=0.
- The same wave shape as sine, shifted by a quarter-cycle.