Function library
Sine Function
f(x) = sin(x)
The classic wave of trigonometry: a smooth oscillation that repeats every 2π.
The sine function is one of the most recognizable curves in all of mathematics: a smooth, repeating wave that oscillates forever between −1 and 1. Originally defined through right triangles — the sine of an angle is the ratio of the opposite side to the hypotenuse — it extends naturally to all real numbers by measuring angles in radians around the unit circle. On the unit circle, sin(x) is simply the y-coordinate of the point reached after rotating x radians from the positive x-axis.
Because it repeats every 2π radians, sine is the prototype of every periodic phenomenon: alternating current, sound waves, light, tides, and the vibration of a guitar string can all be described with sine waves of different frequencies and amplitudes. Type sin(x) into the graphing calculator to trace the wave yourself, then shift it, stretch it, and combine it with other expressions to see how real-world oscillations are built from this one curve.
Computed properties
Calculated by the graphing calculator's own math engine at build time for sin(x).
- Roots in [−10, 10]
- -9.4248, -6.2832, -3.1416, 0.0000, 3.1416, 6.2832, 9.4248
- y-intercept f(0)
- 0.0000
- Local extrema in [−10, 10]
-
- local minimum at (-7.8540, -1.0000)
- local maximum at (-4.7124, 1.0000)
- local minimum at (-1.5708, -1.0000)
- local maximum at (1.5708, 1.0000)
- local minimum at (4.7124, -1.0000)
- local maximum at (7.8540, 1.0000)
- Sample values
- f(-2) = -0.9093 · f(-1) = -0.8415 · f(0) = 0 · f(1) = 0.8415 · f(2) = 0.9093
- Derivative f′(1)
- 0.5403
- Integral ∫₀¹ f(x) dx
- 0.4597
Key facts
- Domain: all real numbers; range: −1 ≤ sin(x) ≤ 1.
- Period 2π: sin(x + 2π) = sin(x) for every x.
- Odd function: sin(−x) = −sin(x); the graph has 180° rotational symmetry about the origin.
- Zeros at x = nπ; maxima of 1 at x = π/2 + 2πn; minima of −1 at x = 3π/2 + 2πn.
- y-intercept at (0, 0); derivative is cos(x); an antiderivative is −cos(x).
What sine is
The unit-circle definition is what lets sine accept any real input, not just angles in a triangle. Starting at (1, 0) and moving counterclockwise around the circle of radius 1, each angle x lands on a point whose height above the x-axis is sin(x). After a full turn of 2π radians you return to where you started, which is why the graph repeats — the function’s history is written into the geometry of the circle.
That geometry also explains the wave’s symmetry: sine is an odd function, meaning sin(−x) = −sin(x), so the left half of the graph is the right half rotated 180° about the origin. The graph crosses the x-axis at every multiple of π, reaches its peak of 1 at π/2 plus each full turn, and bottoms out at −1 at 3π/2 plus each full turn.
Amplitude, period, and phase
Three numbers describe any sine wave: amplitude, period, and phase. The amplitude is the wave’s height — for plain sin(x) it is 1, the distance from the midline y = 0 to each peak. Multiplying by a constant, as in 3*sin(x), stretches the wave vertically without changing its shape, which is how louder sounds and stronger signals are modeled.
The period is the horizontal length of one full cycle: 2π for sin(x). Writing sin(2*x) squeezes two full waves into the same span, halving the period to π and doubling the frequency — the pitch of a note an octave higher. Adding a phase shift, sin(x − π/2), slides the whole wave sideways, which is why cosine is secretly a shifted sine: cos(x) = sin(x + π/2).
Where sine appears
Sine waves are the building blocks of signal processing. Any repeating signal — a musical tone, a radio broadcast, the 50 or 60 Hz hum of mains electricity — can be decomposed into a sum of sine waves of different frequencies, a fact known as Fourier analysis. When two sine waves of nearly equal frequency overlap, they interfere to produce beats, the throbbing effect you hear when two slightly out-of-tune instruments play together.
Beyond signals, sine governs simple harmonic motion: the back-and-forth of a mass on a spring, the swing of a small pendulum, and the up-and-down of a floating buoy all follow sinusoidal curves in time. In geometry and physics, sine projects a rotating quantity onto an axis — the vertical position of a Ferris wheel cabin over time traces exactly sin(x).
Frequently asked questions
Why is the sine graph a wave?
Because sine measures height on the unit circle as you rotate. Going around the circle makes the height rise and fall smoothly and repeat every full turn, so the graph of height versus angle is a wave. The wave is smooth because rotation is continuous — there are no corners or jumps.
What is the difference between sine and cosine?
They are the same wave shifted sideways: cos(x) = sin(x + π/2). On the unit circle, cosine is the x-coordinate while sine is the y-coordinate. Cosine starts at its maximum, cos(0) = 1, while sine starts at zero, sin(0) = 0.
Does sin(x) ever exceed 1?
No — for real x, |sin(x)| ≤ 1 always. On the unit circle, the y-coordinate can never be larger in magnitude than the radius, which is 1. (Sine of complex numbers can exceed 1 in magnitude, but the calculator’s real-valued graph stays within [−1, 1].)