Sine

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For other uses, see Sine (disambiguation).

Sine function

Basic features
Parity	odd
Domain	(−∞,∞)
Codomain	[−1,1]
Period	2π
Specific values
At zero	0
Maxima	((2k + ½)π, 1)
Minima	((2k − ½)π, −1)
Specific features
Root	kπ
Critical point	kπ − π/2
Inflection point	kπ
Fixed point	0
Variable k is an integer. Domain and codomain are given only for real (not complex) numbers.

$\sin \alpha = \frac {\textrm{opposite}} {\textrm{hypotenuse}}$
For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.

The sine function graphed on the Cartesian plane. In this graph, the angle x is given in radians (π = 180°).

The sine and cosine functions are related in multiple ways. The derivative of $\sin(x)$ is $\cos(x)$ . Also they are out of phase by 90°: $\sin(\pi/2 - x)$ = $\cos(x)$ . And for a given angle, cos and sin give the respective x, y coordinates on a unit circle.

In mathematics, the sine function is a trigonometric function of an angle. The sine of an angle is defined in the context of a right triangle: it is the ratio of the length of the side opposite that angle to the length of the hypotenuse.

Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.

The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.^[1] The word "sine" comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.^[2]

Right-angled triangle definition [

For any similar triangle the ratio of the length of the sides remains the same. For example, if the hypotenuse is twice as long, so are the other sides. Therefore respective trigonometric functions, depending only on the size of the angle, express those ratios: between the hypotenuse and the "opposite" side to an angle A in question (see illustration) in the case of sine function; or between the hypotenuse and the "adjacent" side (cosine) or between the "opposite" and the "adjacent" side (tangent), etc.

To define the trigonometric functions for an acute angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows:

The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.
The adjacent side is the side that is in contact with (adjacent to) both the angle we are interested in (angle A) and the right angle, in this case side b.

In ordinary Euclidean geometry, according to the triangle postulate the inside angles of every triangle total 180° (π radians). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be greater than 0° and less than 90°. The following definition applies to such angles.

The angle A (having measure α) is the angle between the hypotenuse and the adjacent line.

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

$\sin \alpha = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {o} {h}.$

This ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar.

Relation to slope [

Main article: Slope

The trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to some horizontal line.

When the length of the line segment is 1, sine takes an angle and tells the rise
Sine takes an angle and tells the rise per unit length of the line segment.
Rise is equal to sin θ multiplied by the length of the line segment

In contrast, cosine is used for the telling the run from the angle; and tangent is used for telling the slope from the angle. Arctan is used for telling the angle from the slope.

The line segment is the equivalent of the hypotenuse in the right-triangle, and when it has a length of 1 it is also equivalent to the radius of the unit circle.

Relation to the unit circle [

In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.

Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The point's distance from the origin is always 1.

Unlike the definitions with the right or left triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function.

The unit circle.

Illustration of a unit circle. The radius has a length of 1. The variable t is an angle measure.

Point P(x,y) on the circle of unit radius at an obtuse angle θ > π/2

Animation showing the graphing process of y = sin x (where x is the angle in radians) using a unit circle. The blue arc around the unit circle (in green) and the blue line at right have the same length, equal to the angle in radians.

Identities [

Exact identities (using radians):

These apply for all values of $\theta$ .

$\begin{align} \sin \theta & = \cos \left(\frac{\pi}{2} - \theta \right) \\ & = \frac{1}{\csc \theta} \end{align}$

Reciprocal [

The reciprocal of sine is cosecant, i.e. the reciprocal of sin(A) is csc(A), or cosec(A). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:

$\csc A = \frac {1}{\sin A} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}.$

Inverse [

The usual principal values of the arcsin(x) function graphed on the cartesian plane. Arcsin is the inverse of sin.

The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin⁻¹). As sine is non-injective, it is not an exact inverse function but a partial inverse function. For example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0 etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value.

$\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \sin^{-1} \left( \frac {a} {h} \right).$

k is some integer:

$\begin{align} \sin y = x \ \Leftrightarrow\ & y = \arcsin x + 2k\pi , \text{ or }\\ & y = \pi - \arcsin x + 2k\pi \end{align}$

Or in one equation:

$\sin y = x \ \Leftrightarrow\ y = (-1)^k \arcsin x + k\pi$

Arcsin satisfies:

$\sin(\arcsin x) = x\!$

and

$\arcsin(\sin \theta) = \theta\quad\text{for }-\pi/2 \leq \theta \leq \pi/2.$

Calculus [

For the sine function:

$f(x) = \sin x \,$

The derivative is:

$f'(x) = \cos x \,$

The antiderivative is:

$\int f(x)\,dx = -\cos x + C$

C denotes the constant of integration.

Other trigonometric functions [

The four quadrants of a Cartesian coordinate system.

It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the sign function).

Sine in terms of the other common trigonometric functions:

	f θ	Using plus/minus (±)					Using sign function (sgn)
		f θ =	± per Quadrant				f θ =
		f θ =	I	II	III	IV	f θ =
cos	$\sin \theta$	$= \pm\sqrt{1 - \cos^2 \theta}$	+	+	-	-	$= \sgn( \cos (\theta - \frac{\pi}{2})) \sqrt{1 - \cos^2\theta}$
cos	$\cos \theta$	$= \pm\sqrt{1 - \sin^2\theta}$	+	-	-	+	$= \sgn( \sin (\theta+ \frac{\pi}{2})) \sqrt{1 - \sin^2\theta}$
cot	$\sin \theta$	$= \pm\frac{1}{\sqrt{1 + \cot^2 \theta}}$	+	+	-	-	$= \sgn( \cot( \frac{\theta}{2})) \frac{1}{\sqrt{1 + \cot^2 \theta}}$
cot	$\cot \theta$	$= \pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}$	+	-	-	+	$= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}$
tan	$\sin \theta$	$= \pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}$	+	-	-	+	$= \sgn( \tan(\frac{2\theta + \pi}{4})) \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}$
tan	$\tan \theta$	$= \pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}$	+	-	-	+	$= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}$
sec	$\sin \theta$	$= \pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}$	+	-	+	-	$= \sgn( \sec ( \frac{4 \theta - \pi}{2})) \frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}$
sec	$\sec \theta$	$= \pm\frac{1}{\sqrt{1 - \sin^2 \theta}}$	+	-	-	+	$= \sgn( \sin (\theta+ \frac{\pi}{2})) \frac{1}{\sqrt{1 - \sin^2 \theta}}$

Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant.

The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity:

$\cos^2\theta + \sin^2\theta = 1\!$

where sin²x means (sin(x))².

Properties relating to the quadrants [

Over the four quadrants of the sine function is as follows.

Quadrant	Degrees	Radians	Value	Sign	Monotony	Convexity
1st Quadrant	$0^\circ<x<90^\circ$	$0<x< \pi/2$	$0<\sin x<1$	$+$	increasing	concave
2nd Quadrant	$90^\circ<x<180^\circ$	$\pi/2<x<\pi$	$0<\sin x<1$	$+$	decreasing	concave
3rd Quadrant	$180^\circ<x<270^\circ$	$\pi<x<3\pi/2$	$-1<\sin x<0$	$-$	decreasing	convex
4th Quadrant	$270^\circ<x<360^\circ$	$3\pi/2<x<2\pi$	$-1<\sin x<0$	$-$	increasing	convex

Points between the quadrants. k is an integer.

The quadrants of the unit circle and of sin x, using the Cartesian coordinate system.

Degrees	Radians 0 ≤ x < 2π	Radians	sin x	Point type
0°	0	$2 \pi k$	0	Root, Inflection
90°	$\pi/2$	$2 \pi k + \pi/2$	1	Maxima
180°	$\pi$	$2 \pi k - \pi$	0	Root, Inflection
270°	$3\pi/2$	$2 \pi k - \pi/2$	-1	Minima

For arguments outside those in the table, get the value using the fact the sine function has a period of 360° (or 2π rad): $\sin(\alpha + 360^\circ) = \sin(\alpha)$ , or use $\sin(\alpha + 180^\circ) = -\sin(\alpha)$ . Or use $\cos(x)= \frac{e^{ix}+e^{-ix}}{2}$ and $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$ . For complement of sine, we have $\sin(180^\circ-\alpha) = \sin(\alpha)$ .

Series definition [

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

This animation shows how including more and more terms in the partial sum of its Taylor series gradually builds up a sine curve.

Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine, and that the derivative of cosine is the negative of sine.

Using the reflection from the calculated geometric derivation of the sine is with the 4n + k-th derivative at the point 0:

$\sin^{(4n+k)}(0)=\begin{cases} 0 & \text{when } k=0 \\ 1 & \text{when } k=1 \\ 0 & \text{when } k=2 \\ -1 & \text{when } k=3 \end{cases}$

This gives the following Taylor series expansion at x = 0. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x (where x is the angle in radians) :^[3]

$\begin{align} \sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[8pt] & = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt] \end{align}$

If x were expressed in degrees then the series would contain messy factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx /180, so

$\begin{align} \sin x_\mathrm{deg} & = \sin y_\mathrm{rad} \\ & = \frac{\pi}{180} x - \left (\frac{\pi}{180} \right )^3\ \frac{x^3}{3!} + \left (\frac{\pi}{180} \right )^5\ \frac{x^5}{5!} - \left (\frac{\pi}{180} \right )^7\ \frac{x^7}{7!} + \cdots . \end{align}$

The series formulas for the sine and cosine are uniquely determined, up to the choice of unit for angles, by the requirements that

$\begin{align} \sin 0 = 0 & \text{ and } \sin{2x} = 2 \sin x \cos x \\ \cos^2 x + \sin^2 x = 1 & \text{ and } \cos{2x} = \cos^2 x - \sin^2 x \\ \end{align}$

The radian is the unit that leads to the expansion with leading coefficient 1 for the sine and is determined by the additional requirement that

$\sin x \approx x \text{ when } x \approx 0.$

The coefficients for both the sine and cosine series may therefore be derived by substituting their expansions into the pythagorean and double angle identities, taking the leading coefficient for the sine to be 1, and matching the remaining coefficients.

In general, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians.

A similar series is Gregory's series for arctan, which is obtained by omitting the factorials in the denominator.

Continued fraction [

The sine function can also be represented as a generalized continued fraction:

$\sin x = \cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 + \cfrac{2\cdot3 x^2}{4\cdot5-x^2 + \cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}.$

The continued fraction representation expresses the real number values, both rational and irrational, of the sine function.

Fixed point [

The fixed point iteration x_n+1 = sin x_n with initial value x₀ = 2 converges to 0.

Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is sin(0) = 0.

Law of sines [

Main article: Law of sines

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}.$

This is equivalent to the equality of the first three expressions below:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,$

where R is the triangle's circumradius.

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.