Graphing the Cosine Function

Learn how to graph the cosine function, understand its key features, and explore transformations.

Advanced25 minLesson
Printable worksheet

Definition

The cosine function is written as y=cos⁡(x)y = \cos(x) and creates a smooth, wave-like curve called a sinusoid.
**Key features of y=cos⁡(x)y = \cos(x):**
PropertyValue
Amplitude11
Period2π2\pi (about 6.286.28)
Maximum11 at x=0,2π,4π,…x = 0, 2\pi, 4\pi, \ldots
Minimum−1-1 at x=π,3π,5π,…x = \pi, 3\pi, 5\pi, \ldots
Zerosx=π2,3π2,5π2,…x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots
The cosine graph starts at its maximum when x=0x = 0:
cos⁡(0)=1\cos(0) = 1
This is the key difference from sine, which starts at zero.

Try it now

What is the value of cos⁡(0)\cos(0)?

Worked Examples

Graph one complete cycle of y=cos⁡(x)y = \cos(x) from x=0x = 0 to x=2πx = 2\pi.
1

Find the starting point

cos⁡(0)=1\cos(0) = 1 → Point: (0,1)(0, 1)
2

Find the first zero

cos⁡(π2)=0\cos\left(\frac{\pi}{2}\right) = 0 → Point: (π2,0)\left(\frac{\pi}{2}, 0\right)
3

Find the minimum

cos⁡(π)=−1\cos(\pi) = -1 → Point: (π,−1)(\pi, -1)
4

Find the second zero

cos⁡(3π2)=0\cos\left(\frac{3\pi}{2}\right) = 0 → Point: (3π2,0)\left(\frac{3\pi}{2}, 0\right)
5

Find the endpoint (back to max)

cos⁡(2π)=1\cos(2\pi) = 1 → Point: (2π,1)(2\pi, 1)

Common Mistakes

Confusing sine and cosine starting points

Why it's wrong: Sine starts at 00 (going up), while cosine starts at its maximum 11. Both are sinusoids but with different starting positions.

Correct: Remember: cos⁡(0)=1\cos(0) = 1 (starts at max), sin⁡(0)=0\sin(0) = 0 (starts at zero)

Getting phase shift direction wrong

Why it's wrong: The equation y=cos⁡(x−C)y = \cos(x - C) shifts RIGHT by CC, not left. The minus sign is counterintuitive.

Correct: y=cos⁡(x−C)y = \cos(x - C) shifts RIGHT; y=cos⁡(x+C)y = \cos(x + C) shifts LEFT

Forgetting to divide 2π2\pi by BB for period

Why it's wrong: Students sometimes think y=cos⁡(2x)y = \cos(2x) has period 22, but period =2πB= \frac{2\pi}{B}.

Correct: For y=cos⁡(Bx)y = \cos(Bx), period =2πB= \frac{2\pi}{B}, not BB

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Graph Sine and Cosine Functions | Step-by-Step Examples

by Understand The Math

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Interactive Visual

Function graph

-2π-π0π2π-4-2024xy(-4.71, 0)(-1.57, 0)(1.57, 0)(4.71, 0)(-3.14, -1)(0, 1)(3.14, -1)

Tap or hover over the graph to read off values.

  • Amplitude = 1
  • Period = 2π
  • Phase shift = 0
  • Midline y = 0
y = A·cos(B(x − C)) + D
  • Zero (x-intercept): (-4.71, 0)
  • Zero (x-intercept): (-1.57, 0)
  • Zero (x-intercept): (1.57, 0)
  • Zero (x-intercept): (4.71, 0)
  • Minimum: (-3.14, -1)
  • Maximum: (0, 1)
  • Minimum: (3.14, -1)

Unit Circle

Degrees

0°

Radians

0

sin(θ)

0

cos(θ)

1

tan(θ)

0

Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Interactive Sandbox

Interactive Grapher

-3π-2π-π0π2π3π-4-2024xy

Tap or hover over the graph to read off values.

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the value of cos⁡(0)\cos(0)?

Why It Matters

The cosine function models countless real-world phenomena:
  • Sound waves: The pressure variations in sound follow cosine patterns
  • Alternating current (AC): Electrical current in your home oscillates like a cosine wave
  • Seasonal temperatures: Average temperatures throughout the year follow a cosine curve
  • Pendulum motion: The horizontal position of a swinging pendulum follows x=Acos⁡(ωt)x = A\cos(\omega t)
  • Light waves: Electromagnetic radiation can be modeled using cosine functions
Understanding cosine graphs helps you analyze any periodic phenomenon!

Real World Applications

Modeling Daily Temperature

Daily temperature follows a cosine pattern, with maximum temperature in the afternoon and minimum at night.

Example:

If the average temperature is 20 degrees Celsius with a 10 degree variation, and maximum occurs at 3 PM, the model is: T(t)=20+10cos⁡(π12(t−15))T(t) = 20 + 10\cos\left(\frac{\pi}{12}(t - 15)\right) where tt is hours after midnight.

1Try It Yourself

A city has average temperature 25 degrees Celsius with variation of 8 degrees. Maximum is at 2 PM (14:00).

What is the temperature at midnight (t = 0)?

Step 1: Write the mathematical expression

Use T(t)=25+8cos⁡(π12(t−14))T(t) = 25 + 8\cos\left(\frac{\pi}{12}(t - 14)\right) and find T(0)T(0):

Sound and Music

Musical notes are produced by sound waves that follow sinusoidal patterns. The cosine function models the pressure variation.

Example:

The note A4 (440 Hz) can be modeled as P(t)=Acos⁡(880πt)P(t) = A\cos(880\pi t) where tt is time in seconds and AA is amplitude.

2Try It Yourself

A sound wave is modeled by P(t)=2cos⁡(1000πt)P(t) = 2\cos(1000\pi t).

What is the frequency of this sound wave?

Step 1: Write the mathematical expression

Use the formula: frequency =B2π= \frac{B}{2\pi} where B=1000πB = 1000\pi:

Key Takeaways

  • 1The cosine function y=cos⁡(x)y = \cos(x) creates a wave that starts at its maximum when x=0x = 0
  • 2Basic properties: amplitude =1= 1, period =2π= 2\pi, range =[−1,1]= [-1, 1]
  • 3For y=Acos⁡(Bx−C)+Dy = A\cos(Bx - C) + D: amplitude =∣A∣= |A|, period =2πB= \frac{2\pi}{B}, phase shift =CB= \frac{C}{B}, vertical shift =D= D
  • 4Cosine is a horizontal shift of sine: cos⁡(x)=sin⁡(x+π2)\cos(x) = \sin\left(x + \frac{\pi}{2}\right)

Frequently Asked Questions

Cosine and sine have the same shape but different starting points. Cosine starts at its maximum (cos⁡(0)=1\cos(0) = 1), while sine starts at zero (sin⁡(0)=0\sin(0) = 0). Mathematically, cos⁡(x)=sin⁡(x+π2)\cos(x) = \sin\left(x + \frac{\pi}{2}\right).
Cosine and sine have the same shape but different starting points. Cosine starts at its maximum (cos⁡(0)=1\cos(0) = 1), while sine starts at zero (sin⁡(0)=0\sin(0) = 0). Mathematically, cos⁡(x)=sin⁡(x+π2)\cos(x) = \sin\left(x + \frac{\pi}{2}\right).
The period is 2π2\pi because cosine is defined using the unit circle, and going around the circle once (360 degrees or 2π2\pi radians) brings you back to the starting point.
A negative amplitude like y=−2cos⁡(x)y = -2\cos(x) reflects the graph across the xx-axis. The wave is flipped upside down: it starts at −2-2 (minimum) instead of 22 (maximum).

Glossary

Amplitude
The height from the midline to the maximum (or minimum). For y=Acos⁡(x)y = A\cos(x), amplitude =∣A∣= |A|.
Period
The horizontal length of one complete cycle. For y=cos⁡(Bx)y = \cos(Bx), period =2πB= \frac{2\pi}{B}.
Phase shift
A horizontal translation of the graph. For y=cos⁡(x−C)y = \cos(x - C), the shift is CC units to the right.
Vertical shift
Moving the entire graph up or down. For y=cos⁡(x)+Dy = \cos(x) + D, the midline becomes y=Dy = D.
Sinusoid
A wave-shaped curve like sine or cosine.
Today's challengeOne problem, three tries

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