Graphing the Sine Function

Learn to graph the sine function and understand its key properties like amplitude, period, and phase.

Advanced25 minLesson
Printable worksheet

Definition

The sine function y=sin⁡(x)y = \sin(x) creates a smooth, wave-like curve called a sinusoidal wave. It oscillates between −1-1 and 11 and repeats every 2π2\pi radians (360 degrees).
**Key properties of y=sin⁡(x)y = \sin(x):**
PropertyValue
Amplitude11 (height from center to peak)
Period2π2\pi (one complete cycle)
DomainAll real numbers
Range[−1,1][-1, 1]
Zerosx=nπx = n\pi where nn is any integer
**Critical points in one period [0,2π][0, 2\pi]:**
sin⁡(0)=0(starts at origin)sin⁡(π2)=1(maximum)sin⁡(π)=0(crosses x-axis)sin⁡(3π2)=−1(minimum)sin⁡(2π)=0(cycle complete)\begin{aligned} \sin(0) &= 0 \quad \text{(starts at origin)} \\ \sin\left(\frac{\pi}{2}\right) &= 1 \quad \text{(maximum)} \\ \sin(\pi) &= 0 \quad \text{(crosses x-axis)} \\ \sin\left(\frac{3\pi}{2}\right) &= -1 \quad \text{(minimum)} \\ \sin(2\pi) &= 0 \quad \text{(cycle complete)} \end{aligned}

Try it now

What is the amplitude of y=sin⁡(x)y = \sin(x)?

Worked Examples

Graph y=sin⁡(x)y = \sin(x) for 0≤x≤2π0 \leq x \leq 2\pi
1

Create a table of key values

Identify the five key points: x=0,π2,π,3π2,2πx = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi → Key x-values identified
2

Calculate the y-values

sin⁡(0)=0\sin(0)=0, sin⁡(π2)=1\sin(\frac{\pi}{2})=1, sin⁡(π)=0\sin(\pi)=0, sin⁡(3π2)=−1\sin(\frac{3\pi}{2})=-1, sin⁡(2π)=0\sin(2\pi)=0 → Points: (0,0)(0,0), (π2,1)(\frac{\pi}{2},1), (π,0)(\pi,0), (3π2,−1)(\frac{3\pi}{2},-1), (2π,0)(2\pi,0)
3

Plot the points

Mark each point on the coordinate plane → Five points plotted
4

Connect with a smooth curve

Draw a smooth wave connecting all points → One complete sine wave

Common Mistakes

Confusing degrees and radians on the x-axis

Why it's wrong: The standard sine graph uses radians. π≈3.14\pi \approx 3.14 radians equals 180 degrees.

Correct: Always check if your calculator and graph are in the same mode. For calculus and higher math, radians are standard.

Getting the phase shift direction wrong

Why it's wrong: In y=sin⁡(x−c)y = \sin(x - c), the shift is to the RIGHT, not left. The minus inside creates opposite behavior.

Correct: y=sin⁡(x−π2)y = \sin(x - \frac{\pi}{2}) shifts RIGHT by π2\frac{\pi}{2}. y=sin⁡(x+π2)y = \sin(x + \frac{\pi}{2}) shifts LEFT by π2\frac{\pi}{2}.

Confusing amplitude with range

Why it's wrong: Amplitude is the distance from the midline to the peak, not the total height.

Correct: For y=3sin⁡(x)y = 3\sin(x), amplitude is 3, but the total height (range) is 6 (from −3-3 to 33).

Forgetting that the coefficient of xx affects period, not amplitude

Why it's wrong: In y=sin⁡(Bx)y = \sin(Bx), BB compresses or stretches horizontally, changing the period.

Correct: Period =2π∣B∣= \frac{2\pi}{|B|}. Larger BB means shorter period (faster oscillation).

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(-6.28, 0)(-3.14, 0)(0, 0)(3.14, 0)(6.28, 0)(-4.71, 1)(-1.57, -1)(1.57, 1)(4.71, -1)

Tap or hover over the graph to read off values.

  • Amplitude = 1
  • Period = 2π
  • Phase shift = 0
  • Midline y = 0
y = A·sin(B(x − C)) + D
  • Zero (x-intercept): (-6.28, 0)
  • Zero (x-intercept): (-3.14, 0)
  • Zero (x-intercept): (0, 0)
  • Zero (x-intercept): (3.14, 0)
  • Zero (x-intercept): (6.28, 0)
  • Maximum: (-4.71, 1)
  • Minimum: (-1.57, -1)
  • Maximum: (1.57, 1)
  • Minimum: (4.71, -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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sin(x)

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the amplitude of y=sin⁡(x)y = \sin(x)?

Why It Matters

The sine function is everywhere in the physical world:
  • Sound waves: Music and speech travel as sinusoidal pressure waves
  • Electricity: Alternating current (AC) follows a sine wave pattern
  • Ocean tides: The rise and fall of tides can be modeled with sine functions
  • Pendulums: A swinging pendulum traces a sine curve over time
  • Seasons: Temperature variations throughout the year follow a sinusoidal pattern
Engineers, physicists, and musicians all rely on understanding sine waves to design everything from bridges to synthesizers.

Real World Applications

Sound Waves and Music

Pure musical tones are sine waves. A tuning fork vibrates at 440 Hz (A note), creating a sine wave in air pressure.

Example:

The equation y=sin⁡(880πt)y = \sin(880\pi t) models a 440 Hz sound wave, where tt is time in seconds.

1Try It Yourself

A guitar string produces a note at 330 Hz.

What is the period of this sound wave in seconds?

Step 1: Write the mathematical expression

Use Period =1frequency= \frac{1}{\text{frequency}}:

Electrical Engineering - AC Current

Household electricity uses alternating current that follows a sine wave pattern.

Example:

European outlets provide 230V at 50 Hz: V(t)=325sin⁡(100πt)V(t) = 325\sin(100\pi t) where the peak voltage is about 325V.

2Try It Yourself

US electricity operates at 60 Hz with peak voltage of 170V.

Write the equation for US household voltage.

Step 1: Write the mathematical expression

Use V(t)=Asin⁡(2πf⋅t)V(t) = A\sin(2\pi f \cdot t) where AA is peak voltage and ff is frequency:

Ocean Tides

Tidal heights follow a roughly sinusoidal pattern due to the moon's gravitational pull.

Example:

If high tide is at 6 AM with height 3 meters and low tide is at 12 PM with height 1 meter, the tide can be modeled with a sine function.

3Try It Yourself

A harbor has a high tide of 4 meters at midnight and a low tide of 0 meters 6 hours later.

What is the period of this tidal function, in hours?

Step 1: Write the mathematical expression

The period is twice the time from high tide to low tide:

Key Takeaways

  • 1The sine function y=sin⁡(x)y = \sin(x) creates a smooth wave oscillating between −1-1 and 11
  • 2Key points occur at x=0,π2,π,3π2,2πx = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi (values: 0,1,0,−1,00, 1, 0, -1, 0)
  • 3The period of the basic sine function is 2π2\pi (one complete cycle)
  • 4In y=Asin⁡(Bx−C)+Dy = A\sin(Bx - C) + D: ∣A∣|A| = amplitude, 2π∣B∣\frac{2\pi}{|B|} = period, CB\frac{C}{B} = phase shift, DD = vertical shift
  • 5Amplitude is the distance from the midline to the peak, not the total height

Frequently Asked Questions

On the unit circle, sine represents the y-coordinate. At angle 0 (pointing right on the x-axis), the y-coordinate is 0. This is why sin⁡(0)=0\sin(0) = 0.
On the unit circle, sine represents the y-coordinate. At angle 0 (pointing right on the x-axis), the y-coordinate is 0. This is why sin⁡(0)=0\sin(0) = 0.
The cosine graph is the same shape as sine, but shifted left by π2\frac{\pi}{2}. In fact, cos⁡(x)=sin⁡(x+π2)\cos(x) = \sin(x + \frac{\pi}{2}). Cosine starts at its maximum (1) while sine starts at 0.
A negative coefficient reflects the graph across the x-axis. The amplitude is still ∣−2∣=2|-2| = 2. The wave that normally goes up first now goes down first: the maxima become minima and the minima become maxima.

Glossary

Amplitude
The distance from the midline to the maximum (or minimum) of the wave. For y=Asin⁡(x)y = A\sin(x), amplitude is ∣A∣|A|.
Period
The horizontal length of one complete cycle. For y=sin⁡(Bx)y = \sin(Bx), period is 2π∣B∣\frac{2\pi}{|B|}.
Phase shift
A horizontal translation of the graph. In y=sin⁡(x−C)y = \sin(x - C), the phase shift is CC units to the right.
Midline
The horizontal line halfway between the maximum and minimum values. For y=sin⁡(x)+Dy = \sin(x) + D, the midline is y=Dy = D.
Sinusoidal
Having the shape of a sine curve; any function that can be written as a transformed sine or cosine function.

Formula Card

General Form

y=Asin⁡(Bx−C)+Dy = A\sin(Bx - C) + D

Complete sine transformation formula

Amplitude

∣A∣|A|

Height from midline to peak

Period

2π∣B∣\frac{2\pi}{|B|}

Length of one complete cycle

Phase Shift

CB\frac{C}{B}

Horizontal shift (right if positive); in the form $y = A\sin(B(x - h)) + D$ the shift is simply $h$

Vertical Shift

DD

Moves the midline up or down

Key Points

(0,0),(π2,1),(π,0),(3π2,−1),(2π,0)(0,0), (\frac{\pi}{2},1), (\pi,0), (\frac{3\pi}{2},-1), (2\pi,0)

Five critical points for one cycle

Today's challengeOne problem, three tries

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