y = \sin x

Graph of the Sine Function $y = \sin x$

$y = \sin x$ is the $y$ -coordinate of the point at angle $x$ , measured in radians, on the unit circle. As the angle grows the point travels around the circle and its height rises and falls; stretching that motion out along the horizontal axis traces this sine curve¹.

Domain and range

The domain is all real numbers. Since the value is the height of a point on the unit circle, $-1 \leq \sin x \leq 1$ , so the range is the interval $[-1, 1]$ .

The domain is all real numbers
The range is $-1 \leq y \leq 1$
The maximum is $1$ and the minimum is $-1$
The amplitude is $1$ and the period is $2\pi$

Periodicity

Going once around the circle returns the point to where it started, so $\sin(x + 2\pi) = \sin x$ . The period is $2\pi$ , and the same wave repeats endlessly in both directions.

Symmetry

Since $\sin(-x) = -\sin x$ , it is an odd function whose graph has point symmetry about the origin.

Monotonicity and extrema

The derivative is $y' = \cos x$ , so the function increases where $\cos x > 0$ and decreases where $\cos x < 0$ . It attains its maximum $1$ at $x = \dfrac{\pi}{2} + 2n\pi$ and its minimum $-1$ at $x = \dfrac{3\pi}{2} + 2n\pi$ .

The second derivative $y'' = -\sin x$ changes sign at every zero $x = n\pi$ , so the inflection points sit at $x = n\pi$ , where the tangent has slope $\pm 1$ . Zeros and inflection points coincide because the function satisfies $y'' = -y$ .

Notable values

$x$	$\sin x$
$0$	$0$
$\dfrac{\pi}{6}$	$\dfrac{1}{2}$
$\dfrac{\pi}{4}$	$\dfrac{\sqrt{2}}{2}$
$\dfrac{\pi}{3}$	$\dfrac{\sqrt{3}}{2}$
$\dfrac{\pi}{2}$	$1$
$\pi$	$0$

The zeros are the solutions of $\sin x = 0$ , namely $x = n\pi$ for integer $n$ , evenly spaced $\pi$ apart.

Approximation near the origin

The tangent at the origin is the line $y = x$ , of slope $1$ , which reflects the following limit.

\lim_{x \to 0} \frac{\sin x}{x} = 1

This is why $\sin x \approx x$ for small angles. The Taylor series converges for every real number.

\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots

That only odd powers appear is how oddness shows up on the side of the series.

The cycle of derivatives

Times differentiated	Derivative
$1$	$\cos x$
$2$	$-\sin x$
$3$	$-\cos x$
$4$	$\sin x$

Four steps bring it back to the start.

Relationships with other functions

Since $\cos x = \sin\left( x + \dfrac{\pi}{2} \right)$ , the cosine is this wave shifted left by $\dfrac{\pi}{2}$ . The identity $\sin^2 x + \cos^2 x = 1$ is just the equation of the unit circle. In Euler's formula $e^{ix} = \cos x + i\sin x$ , the sine supplies the imaginary part.

Applications

Simple harmonic motion, such as a spring or a pendulum
Waves of sound, light and radio
Alternating current and voltage
The basic wave of Fourier analysis

Fourier analysis goes further: any periodic wave can be written as a superposition of sine waves². The sine curve is the basic shape of anything that repeats.

Sine and cosine, Wikipedia
Fourier series, Wikipedia

Graph of the Sine Function y=sin⁡xy = \sin xy=sinx