Right-angled triangle definitions

For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.

To define the sine and cosine of an acute angle α, start with a right triangle that contains an angle of measure α; in the accompanying figure, angle α in triangle ABC is the angle of interest. The three sides of the triangle are named as follows:

• The opposite side is the side opposite to the angle of interest, in this case side a.
• 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 adjacent side is the remaining side, in this case side b. It forms a side of (and is adjacent to) both the angle of interest (angle A) and the right angle.

Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side, divided by the length of the hypotenuse:^[2]

${\displaystyle \sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\qquad \cos(\alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}}$

The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides.^[2]

As stated, the values ${\displaystyle \sin(\alpha )}$ and ${\displaystyle \cos(\alpha )}$ appear to depend on the choice of right triangle containing an angle of measure α. However, this is not the case: all such triangles are similar, and so the ratios are the same for each of them.

Unit circle definitions

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

Unit circle: a circle with radius one

Let a line through the origin intersect the unit circle, making an angle of θ with the positive half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos(θ) and sin(θ), respectively. This definition is consistent with the right-angled triangle definition of sine and cosine when ${\displaystyle 0<\theta <{\frac {\pi }{2}}}$: because the length of the hypotenuse of the unit circle is always 1, ${\displaystyle \sin(\theta )={\frac {\text{opposite}}{\text{hypotenuse}}}={\frac {\text{opposite}}{1}}={\text{opposite}}}$. The length of the opposite side of the triangle is simply the y-coordinate. A similar argument can be made for the cosine function to show that ${\displaystyle \cos(\theta )={\frac {\text{adjacent}}{\text{hypotenuse}}}}$ when ${\displaystyle 0<\theta <{\frac {\pi }{2}}}$, even under the new definition using the unit circle. tan(θ) is then defined as ${\displaystyle {\frac {\sin(\theta )}{\cos(\theta )}}}$, or, equivalently, as the slope of the line segment.

Using the unit circle definition has the advantage that the angle can be extended to any real argument. This can also be achieved by requiring certain symmetries, and that sine be a periodic function.

Complex exponential function definitions

The exponential function ${\displaystyle e^{z}}$ is defined on the entire domain of the complex numbers. The definition of sine and cosine can be extended to all complex numbers via

${\displaystyle \sin z={\frac {e^{iz}-e^{-iz}}{2i}}}$

${\displaystyle \cos z={\frac {e^{iz}+e^{-iz}}{2}}}$

These can be reversed to give Euler's formula

${\displaystyle e^{iz}=\cos z+i\sin z}$

${\displaystyle e^{-iz}=\cos z-i\sin z}$

When plotted on the complex plane, the function ${\displaystyle e^{ix}}$ for real values of ${\displaystyle x}$ traces out the unit circle in the complex plane.

When ${\displaystyle x}$ is a real number, sine and cosine simplify to the imaginary and real parts of ${\displaystyle e^{ix}}$ or ${\displaystyle e^{-ix}}$, as:

${\displaystyle \sin x=\operatorname {Im} (e^{ix})=-\operatorname {Im} (e^{-ix})}$

${\displaystyle \cos x=\operatorname {Re} (e^{ix})=\operatorname {Re} (e^{-ix})}$

When ${\displaystyle z=x+iy}$ for real values ${\displaystyle x}$ and ${\displaystyle y}$, sine and cosine can be expressed in terms of real sines, cosines, and hyperbolic functions as

${\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}}$

Differential equation definition

${\displaystyle (\cos \theta ,\sin \theta )}$ is the solution ${\displaystyle (x(\theta ),y(\theta ))}$ to the two-dimensional system of differential equations ${\displaystyle y'(\theta )=x(\theta )}$ and ${\displaystyle x'(\theta )=-y(\theta )}$ with the initial conditions ${\displaystyle y(0)=0}$ and ${\displaystyle x(0)=1}$. One could interpret the unit circle in the above definitions as defining the phase space trajectory of the differential equation with the given initial conditions.

• 
  Animation demonstrating how the sine function (in red) is graphed from the y-coordinate (red dot) of a point on the unit circle (in green), at an angle of θ. The cosine (in blue) is the x-coordinate.

It can be interpreted as a phase space trajectory of the system of differential equations ${\displaystyle y'(\theta )=x(\theta )}$ and ${\displaystyle x'(\theta )=-y(\theta )}$ starting from the initial conditions ${\displaystyle y(0)=0}$ and ${\displaystyle x(0)=1}$.

Series definitions

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 approaches a sine curve.

The successive derivatives of sine, evaluated at zero, can be used to determine its Taylor series. 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. This means the successive derivatives of sin(x) are cos(x), -sin(x), -cos(x), sin(x), continuing to repeat those four functions. The (4n+k)-th derivative, evaluated at the point 0:

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

where the superscript represents repeated differentiation. This implies 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]

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

Taking the derivative of each term gives the Taylor series for cosine:

${\displaystyle {\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\[8pt]\end{aligned}}}$

Continued fraction definitions

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

${\displaystyle \sin(x)={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}.}$

${\displaystyle \cos(x)={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}.}$

The continued fraction representations can be derived from Euler's continued fraction formula and express the real number values, both rational and irrational, of the sine and cosine functions.

Reciprocals

The reciprocal of sine is cosecant, i.e., the reciprocal of ${\displaystyle \sin {\theta }}$ is ${\displaystyle \csc {\theta }}$. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the length of the hypotenuse to that of the adjacent side.

${\displaystyle \csc {\theta }={\frac {1}{\sin {\theta }}}={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}}$

${\displaystyle \sec {\theta }={\frac {1}{\cos {\theta }}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}}$

Inverses

The usual principal values of the arcsin(x) and arccos(x) functions graphed on the Cartesian plane

The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin⁻¹). The inverse function of cosine is arccosine (arccos, acos, or cos⁻¹). (The superscript of −1 in sin⁻¹ and cos⁻¹ denotes the inverse of a function, not exponentiation.) As sine and cosine are not injective, their inverses are not exact inverse functions, but partial inverse functions. 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. The standard range of principal values for arcsin is from −π/2 to π/2 and the standard range for arccos is from 0 to π.

${\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right).}$

where (for some integer k):

${\displaystyle {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\arccos(x)+2\pi k,{\text{ or }}\\&y=-\arccos(x)+2\pi k\end{aligned}}}$

By definition, arcsin and arccos satisfy the equations:

${\displaystyle \sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x}$

and

${\displaystyle {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq \theta \leq \pi \end{aligned}}}$

Pythagorean trigonometric identity

The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:^[1]

${\displaystyle \cos ^{2}(\theta )+\sin ^{2}(\theta )=1}$

where sin²(x) means (sin(x))².

Double angle formulas

Sine and cosine satisfy the following double angle formulas:

${\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}$

${\displaystyle \cos(2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )=2\cos ^{2}(\theta )-1=1-2\sin ^{2}(\theta )}$

Sine function in blue and sine squared function in red. The X axis is in radians.

The cosine double angle formula implies that sin² and cos² are, themselves, shifted and scaled sine waves. Specifically,^[4]

${\displaystyle \sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}}$

The graph shows both the sine function and the sine squared function, with the sine in blue and sine squared in red. Both graphs have the same shape, but with different ranges of values, and different periods. Sine squared has only positive values, but twice the number of periods.

Derivative and integrals

The derivatives of sine and cosine are:

${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)\qquad {\frac {d}{dx}}\cos(x)=-\sin(x)}$

and their antiderivatives are:

${\displaystyle \int \sin(x)\,dx=-\cos(x)+C}$

${\displaystyle \int \cos(x)\,dx=\sin(x)+C}$

where C denotes the constant of integration.^[1]

Complex arguments

Domain coloring of sin(z) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.

sin(z) as a vector field

Applying the series definition of the sine and cosine to a complex argument, z, gives:

${\displaystyle {\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=-i\sinh \left(iz\right)\\\cos(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{-iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}}$

where sinh and cosh are the hyperbolic sine and cosine. These are entire functions.

It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:

${\displaystyle {\begin{aligned}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)-\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)-i\sin(x)\sinh(y)\\\end{aligned}}}$

Complex graphs

Sine function in the complex plane

real component	imaginary component	magnitude

Arcsine function in the complex plane

real component	imaginary component	magnitude

Etymology

Etymologically, the word sine derives from the Sanskrit word jyā 'bow-string'^[17][18] or more specifically its synonym jīvá (both adopted from Ancient Greek χορδή 'string'^[19]), due to visual similarity between the arc of a circle with its corresponding chord and a bow with its string (see jyā, koti-jyā and utkrama-jyā). This was transliterated in Arabic as jība, which is however meaningless in that language and abbreviated jb (جب). Since Arabic is written without short vowels, jb was interpreted as the homograph jaib, jayb (جيب), which means 'bosom', 'pocket', 'fold'. When the Arabic texts of Al-Battani and al-Khwārizmī were translated into Medieval Latin in the 12th century by Gerard of Cremona, he used the Latin equivalent sinus (which also means 'bay' or 'fold', and more specifically 'the hanging fold of a toga over the breast').^[11][20][21] Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.^[22][23] The English form sine was introduced in the 1590s.^[24]

The word cosine derives from an abbreviation of the Latin complementi sinus 'sine of the complementary angle' as cosinus in Edmund Gunter's Canon triangulorum (1620), which also includes a similar definition of cotangens.^[25][26][27]

Turns based implementations

Some software libraries provide implementations of sine and cosine using the input angle in half-turns, a half-turn being an angle of 180 degrees or ${\displaystyle \pi }$ radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.^[29][30] In MATLAB, OpenCL, R, Julia, CUDA, and ARM, these function are called sinpi and cospi.^{[29][31][30][32][33][34]} For example, sinpi(x) would evaluate to ${\displaystyle \sin(\pi x),}$ where x is expressed in half-turns, and consequently the final input to the function, πx can be interpreted in radians by sin.

The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast, representing ${\displaystyle 2\pi }$, ${\displaystyle \pi }$, and ${\textstyle {\frac {\pi }{2}}}$ in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since irrational numbers cannot be represented with finitely many binary digits.

Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo ${\textstyle {\frac {\pi }{2}}}$ involves inaccuracies in representing ${\textstyle {\frac {\pi }{2}}}$.

For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to 4096 over one complete revolution.^[35] If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to ${\textstyle {\frac {\pi }{2048}}}$ would be incurred.