Graphing the Cosine Function

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

Advanced25 minLesson

Definition

The cosine function is written as

y = \cos(x)

and creates a smooth, wave-like curve called a sinusoid.

**Key features of

y = \cos(x)

:**

Property	Value
Amplitude	$1$
Period	$2\pi$ (about $6.28$ )
Maximum	$1$ at $x = 0, 2\pi, 4\pi, \ldots$
Minimum	$-1$ at $x = \pi, 3\pi, 5\pi, \ldots$
Zeros	$x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots$

The cosine graph starts at its maximum when

x = 0

\cos(0) = 1

This is the key difference from sine, which starts at zero.

Try it now

What is the value of

\cos(0)

Worked Examples

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

Find the starting point

$\cos(0) = 1$ → Point: $(0, 1)$

Find the first zero

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

Find the minimum

$\cos(\pi) = -1$ → Point: $(\pi, -1)$

Find the second zero

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

Find the endpoint (back to max)

$\cos(2\pi) = 1$ → Point: $(2\pi, 1)$

Answer: The five key points are $(0, 1)$ , $\left(\frac{\pi}{2}, 0\right)$ , $(\pi, -1)$ , $\left(\frac{3\pi}{2}, 0\right)$ , and $(2\pi, 1)$ . Connect these with a smooth wave.

Common Mistakes

Confusing sine and cosine starting points

Why it's wrong: Sine starts at $0$ (going up), while cosine starts at its maximum $1$ . Both are sinusoids but with different starting positions.

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

Getting phase shift direction wrong

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

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

Forgetting to divide $2\pi$ by $B$ for period

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

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

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is the value of $\cos(0)$ ?

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 $x = A\cos(\omega t)$
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: $T(t) = 20 + 10\cos\left(\frac{\pi}{12}(t - 15)\right)$ where $t$ 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 $T(t) = 25 + 8\cos\left(\frac{\pi}{12}(t - 14)\right)$ and find $T(0)$ :

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 $P(t) = A\cos(880\pi t)$ where $t$ is time in seconds and $A$ is amplitude.

2Try It Yourself

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

What is the frequency of this sound wave?

Step 1: Write the mathematical expression

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

Key Takeaways

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

Frequently Asked Questions

Cosine and sine have the same shape but different starting points. Cosine starts at its maximum (

\cos(0) = 1

), while sine starts at zero (

\sin(0) = 0

). Mathematically,

\cos(x) = \sin\left(x + \frac{\pi}{2}\right)

Glossary

Amplitude: The height from the midline to the maximum (or minimum). For $y = A\cos(x)$ , amplitude $= |A|$ .
Period: The horizontal length of one complete cycle. For $y = \cos(Bx)$ , period $= \frac{2\pi}{B}$ .
Phase shift: A horizontal translation of the graph. For $y = \cos(x - C)$ , the shift is $C$ units to the right.
Vertical shift: Moving the entire graph up or down. For $y = \cos(x) + D$ , the midline becomes $y = D$ .
Sinusoid: A wave-shaped curve like sine or cosine.