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Sine graph: y = sin x

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The sine graph is a smooth wave that repeats every 2π units, rising to 1 and falling to −1. It passes through the origin, and its shape describes everything from sound waves to the height of a point on a turning wheel.

−6−4−2246−4−2240
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The graph of
y=sin(x)
 for −7 ≤ x ≤ 7. Dots mark its roots, turning points and y-intercept.

Key features

Domain
All real numbers
Range
−1 ≤ y ≤ 1
Period
2π ≈ 6.2832
Amplitude
1
Zeros
x = nπ for every integer n
Maximum
y = 1 at x = π/2 + 2nπ
Minimum
y = −1 at x = 3π/2 + 2nπ
Symmetry
Odd: sin(−x) = −sin x, so the graph is symmetric about the origin
y-intercept
None
Derivative
cos(x)
Antiderivative
−cos(x)+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of sin x looks like

Start at the origin. As x increases from 0 to π/2 the graph climbs to its maximum of 1, then falls back through zero at x = π, reaches its minimum of −1 at x = 3π/2 and returns to zero at x = 2π. That single arch and trough is one full period; the rest of the graph is copies of it, end to end, in both directions.

The graph is steepest where it crosses the x-axis, where its slope is exactly ±1, and flat at the peaks and troughs. That is the derivative cos x at work: cos x is ±1 at the zeros of sin x and 0 at its extremes. The points where the curve changes from bending down to bending up, its inflection points, are exactly those zeros.

Because sin(−x) = −sin x, the graph is symmetric about the origin: turn it half a turn about (0, 0) and it lands on itself. Shifting it left by π/2 gives the cosine graph exactly, since sin(x + π/2) = cos x.

Points of interest

Found numerically to about twelve digits in the window −7 ≤ x ≤ 7, then shown in exact form where one exists.

x-intercepts (roots)
x = −6.2832, x = −3.1416, x = 0, x = 3.1416, x = 6.2832
Local maxima
(−4.7124, 1), (1.5708, 1)
Local minima
(−1.5708, −1), (4.7124, −1)
Inflection points
(−2π, 0), (−π, 0), (0, 0), (π, 0), (2π, 0)

Table of values

A few points to plot, with exact values where they are simple.

Values of sin x
xy = sin x
00
π/61/2
π/4√2/2
π/3√3/2
π/21
2π/3√3/2
π0
3π/2−1
2π0

How to graph sin x by hand

  1. 1
    Mark the x-axis in multiples of π/2: −2π, −3π/2, −π, −π/2, 0, π/2, π, 3π/2, 2π. (π/2 is about 1.57, so on squared paper two squares per unit works well.)
  2. 2
    Plot the five key points of one period: (0, 0), (π/2, 1), (π, 0), (3π/2, −1) and (2π, 0).
  3. 3
    Join them with a smooth curve that is rounded at the peak and trough and crosses the axis at a slant. Straight lines between the points are the most common error.
  4. 4
    Repeat the same shape every 2π in both directions.
  5. 5
    Check: the curve never goes above 1 or below −1, and it passes through the origin.

Transformations

How the graph moves when you change the formula. The original is sin x in blue.

−6−4−2246−4−2240
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  • y = 2sin(x) Amplitude 2: the wave is stretched vertically and now runs from −2 to 2. The zeros do not move.
  • y = sin(2x) Period π: the wave is squeezed horizontally, so two full waves fit where one did. In general sin(bx) has period 2π/b.
  • y = sin(x + pi/2) Shifted left by π/2. This is exactly the cosine graph.

Common mistakes

  • Mixing up degrees and radians. On an axis marked in radians, sin(90) is about 0.894, not 1, because 90 radians is not 90°.
  • Drawing sharp points at the peaks. The sine curve is smooth everywhere; its slope is zero at the top and bottom, not undefined.
  • Taking the period of sin(bx) to be b. It is 2π/b, so a larger b means a shorter period.

Where it shows up

  • Simple harmonic motion: a mass on a spring, or a pendulum swinging through a small angle.
  • Alternating current, where the voltage varies sinusoidally with time (230 V at 50 Hz is about 325 sin(100πt) volts).
  • Sound and light waves, and any periodic signal, which Fourier analysis breaks into sums of sine waves.

Frequently asked questions

What is the period of the sine graph?
The period of y = sin x is 2π (about 6.283) when x is in radians, or 360° in degrees. For y = sin(bx) the period is 2π/|b|.
What is the amplitude of sin x?
The amplitude is 1: the graph rises 1 unit above its midline y = 0 and falls 1 unit below it. For y = A sin x the amplitude is |A|.
Where does sin x equal zero?
At every integer multiple of π: x = …, −2π, −π, 0, π, 2π, … These are the x-intercepts of the graph.
Is sine an odd or even function?
Odd. sin(−x) = −sin x for every x, so the graph has rotational symmetry about the origin.
What is the difference between the sine and cosine graphs?
They are the same wave shifted by a quarter of a period: cos x = sin(x + π/2). Sine is 0 at x = 0, while cosine starts at its maximum value of 1.

Related graphs

Graph of cosine: y = cos x

The sine wave shifted left by π/2; starts at its peak.

Tan graph: y = tan x

Period π, asymptotes at π/2 + nπ, always increasing.

Graph of arcsin x (inverse sine)

Domain [−1, 1], range [−π/2, π/2]; sin x reflected in y = x.

Graph of cosecant: y = csc x

1/sin x: asymptotes at every multiple of π.

Graph of x sin x

Oscillations that grow, trapped between y = ±x.

Graph of sin x / x (the sinc function)

A decaying wave with limit 1 at the origin.

Graph sin x your way

Change the window, add sliders for a, b, h and k, trace the curve with your keyboard, or overlay its derivative.

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