Sin Graph

Sin Graph Calculator

Q: What is the sin graph?

The sin graph is a smooth, wave-like curve of y = sin(x). It oscillates between −1 and 1 with period 2π, crossing zero at x = 0, π, 2π, … It is one of the fundamental trigonometric functions.

Q: What is the period of sin(x)?

The period of sin(x) is 2π (≈ 6.28 radians or 360°). For y = sin(Bx), the period is 2π/B.

Q: How does amplitude affect the sine graph?

Amplitude A in y = A·sin(x) is the wave's height. The graph oscillates between −|A| and +|A|. Example: y = 2 sin(x) has amplitude 2, reaching a maximum of 2 and a minimum of −2.

Q: What is phase shift in sine functions?

Phase shift moves the graph horizontally. For y = A sin(Bx + C), the phase shift is −C/B. Example: y = sin(x − π/2) shifts the graph π/2 units right, making it look like cosine.

Q: Where is sin graph used in real life?

Sine waves appear in sound, music, ocean tides, alternating current, seasonal cycles, light waves, pendulum motion, and springs. Engineers use them to model oscillations across physics, electrical engineering, and signal processing.

Interactive sine wave grapher. Visualize y = A·sin(Bx + C) + D with real-time adjustments for amplitude, period, phase shift, and vertical shift.

100% FreeInteractiveEducational

Sine Graph

y = sin(x)

X-axis pi markers

-4π-7π/2-3π-5π/2-2π-3π/2-π-π/20π/2π3π/2

Y-axis integer ticks

-4-3-2-11234

Graph Controls

Adjust parameters to see how they affect the sine curve

General Sine Function

y = A \sin(Bx + C) + D

Amplitude (A) = 1

Controls the height of the wave (max = A, min = -A). Limited to ±2 to fit graph bounds.

Period (2π/B) = 6.28

Controls how quickly the wave repeats (1 = standard 2π period)

Phase Shift (C) = 0.00

Shifts the wave left (positive C) or right (negative C)

Vertical Shift (D) = 0

Moves the entire wave up (positive D) or down (negative D). Limited to ±2 to fit graph bounds.

Maximum

1.00

Minimum

-1.00

Period

6.28

Midline

y = 0

Preset Examples

Click to see common sine function variations

Understanding the Sine Function

The sine function is one of the fundamental trigonometric functions, representing the y-coordinate of a point on the unit circle.

Key Properties of y = sin(x):

Domain: All real numbers (-∞, +∞)
Range: [-1, 1]
Period: 2π (≈ 6.28)
Amplitude: 1
Zeros: x = nπ where n is any integer
Maximum: 1 at x = π/2 + 2nπ
Minimum: -1 at x = 3π/2 + 2nπ

Odd Function: sin(-x) = -sin(x), meaning the graph is symmetric about the origin.

Parameter Effects

Amplitude (A)

Stretches or compresses vertically. |A| is the distance from midline to peak.

If A < 0, the graph is reflected over the x-axis.

Period (2π/B)

B > 1 compresses horizontally (shorter period)

0 < B < 1 stretches horizontally (longer period)

Phase Shift (-C/B)

Shifts the graph horizontally. Positive C shifts left, negative C shifts right.

Vertical Shift (D)

Moves the entire graph up (D > 0) or down (D < 0).

Frequently Asked Questions

What is the difference between sin and cos graphs?

The cosine graph is the same as the sine graph shifted left by π/2. Mathematically: cos(x) = sin(x + π/2). Both have the same shape, amplitude, and period, just different starting points.

How do I find the equation from a sine graph?

1) Find amplitude: (max - min) / 2 = A
2) Find vertical shift: (max + min) / 2 = D
3) Find period: distance between two maximums, then B = 2π/period
4) Find phase shift: where the first "standard" sine pattern begins

Why is the sine function important?

Sine waves model periodic phenomena in nature: sound waves, light waves, ocean tides, AC electricity, pendulum motion, and more. They're fundamental in physics, engineering, signal processing, and music.

Precalculus Reference

OpenStax Precalculus (Trigonometric Functions)LibreTexts: Graphs of Sine and Cosine

Frequently Asked Questions

What is the sin graph?

The sin graph is a smooth, wave-like curve of y = sin(x). It oscillates between −1 and 1 with period 2π, crossing zero at x = 0, π, 2π, … It is one of the fundamental trigonometric functions.

What is the period of sin(x)?

The period of sin(x) is 2π (≈ 6.28 radians or 360°). For y = sin(Bx), the period is 2π/B.

How does amplitude affect the sine graph?

Amplitude A in y = A·sin(x) is the wave's height. The graph oscillates between −|A| and +|A|. Example: y = 2 sin(x) has amplitude 2, reaching a maximum of 2 and a minimum of −2.

What is phase shift in sine functions?

Phase shift moves the graph horizontally. For y = A sin(Bx + C), the phase shift is −C/B. Example: y = sin(x − π/2) shifts the graph π/2 units right, making it look like cosine.

Where is sin graph used in real life?

Sine waves appear in sound, music, ocean tides, alternating current, seasonal cycles, light waves, pendulum motion, and springs. Engineers use them to model oscillations across physics, electrical engineering, and signal processing.

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