How to Graph the Cosine Function?

Cosine is one of the primary mathematical trigonometric ratios. In this guide, you will learn more about the graph of the cosine function.

How to Graph the Cosine Function?
Tutor-style math help

Graph the Cosine Function: what to notice and how to work it

Trigonometry skill
A cosine graph is a repeating wave that starts at a high point on the parent graph. Mark amplitude, period, midline, and five key points before sketching.

What to notice first

For \(y=A\cos(Bx)+D\), amplitude is \(|A|\), period is \(2\pi/|B|\), and midline is \(y=D\).

Common student mistake

Do not use sine's starting point automatically. Parent cosine starts at \((0,1)\), not \((0,0)\).

Key formulas and cues

\(y=A\cos(Bx)+D\)
\(\text{amplitude}=|A|\)
\(\text{period}=\frac{2\pi}{|B|}\)
\(\text{midline}=y=D\)
amplitude midline

A reliable path

  1. Choose the modelUse a right triangle, the unit circle, or a transformed graph.
  2. Track unitsConvert degrees and radians when needed.
  3. Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Read a cosine rule

Example: \(y=2\cos(4x)+3\)
  1. Amplitude is |2|.
  2. Period is 2pi/4 = pi/2.
  3. Midline is y = 3.
Answer: Amplitude \(2\), period \(\pi/2\), midline \(y=3\).

Place parent cosine points

Example: Graph one cycle of \(y=\cos x\).
  1. Start at (0, 1).
  2. Use quarter-period steps.
  3. The y-values are 1, 0, -1, 0, 1.
Answer: \((0,1),(\pi/2,0),(\pi,-1),(3\pi/2,0),(2\pi,1)\).
Open pop-up practice
Try one before moving on
Try: Find the amplitude and period of \(y=3\cos(6x)\).
Answer: Amplitude \(3\), period \(\pi/3\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

A step-by-step guide to graphing the cosine function

Cosine is one of the primary mathematical trigonometric ratios. The cosine can be defined as the ratio of the length of the base to the length of the hypotenuse in a right triangle.

\(\color{blue}{cos\:x\:=\frac{Adjacent\:Side}{Hypotenuse}=\frac{Base}{Hypotenuse}\:}\)

where \(x\) is the acute angle between the base and the hypotenuse.

A cosine graph or \(cos\) graph is a top-down graph just like a \(sin\) graph. The only difference between a \(sin\) graph and a \(cos\) graph is that a sine graph starts at \(0\) while a cos graph starts at \(90\) (or \(\frac{\pi }{2})\). The \(cos\) plot given below starts at \(1\) and decreases to \(-1\) and then starts increasing again.

The Ultimate Algebra Bundle 2026: From Pre-Algebra to Algebra II

The Ultimate Algebra Bundle 2026 From Pre-Algebra to Algebra II

Original price was: $109.99.Current price is: $54.99.
The Ultimate Elementary School Math Bundle: Grade 3-5

The Ultimate Elementary School Math Bundle Grade 3-5

Original price was: $109.99.Current price is: $54.99.
Effortless Math Team

Effortless Math Team

Related to This Article

Cosine Function Graph of the Cosine Function

More math articles

What people say about "How to Graph the Cosine Function? - Effortless Math"?

No one replied yet.

Leave a Reply

Dallas, Texas
info@EffortlessMath.com

Free math practice in your inbox

Worksheets, practice tests, and study tips for your exam. No spam.

Worksheets & practice pages25,000+
Countries using our resources192+
Practice questions answered8,000,000+
Publications & resources8,000+
Happy students since 20141,000,000+
Free Resources
Popular Test Prep
Math Books
Company & Trust

Effortless Math provides unofficial test prep products for a variety of tests and exams. All trademarks are property of their respective trademark owners.

Effortless Math - © 2026