Sine and cosine#Relationship to complex numbers

File:Trigono sine en2.svg

To define the sine and cosine of an acute angle $\backslash alpha$, start with a right triangle that contains an angle of measure $\backslash alpha$; in the accompanying figure, angle $\backslash alpha$ in a right triangle $ABC$ is the angle of interest. The three sides of the triangle are named as follows:{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27 27]}}

• The opposite side is the side opposite to the angle of interest; in this case, it is $a$.
• The hypotenuse is the side opposite the right angle; in this case, it is $h$. The hypotenuse is always the longest side of a right-angled triangle.
• The adjacent side is the remaining side; in this case, it is $b$. It forms a side of (and is adjacent to) both the angle of interest 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, and the cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse:{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27 27]}}

\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad

\cos(\alpha) = \frac{\text{adjacent}}{\text{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 or equivalently the ratio between the sine and cosine functions. The reciprocal of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be formulated as:{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA27 27]}}

$$\backslash begin\{align\}$$

\tan(\theta) &= \frac{\sin (\theta)}{\cos (\theta)} = \frac{\text{opposite}}{\text{adjacent}}, \\

\cot(\theta) &= \frac{1}{\tan (\theta)} = \frac{\text{adjacent}}{\text{opposite}}, \\

\csc(\theta) &= \frac{1}{\sin (\theta)} = \frac {\text{hypotenuse}}{\text{opposite}}, \\

\sec(\theta) &= \frac{1}{\cos (\theta)} = \frac {\textrm{hypotenuse}} {\textrm{adjacent}}.

\end{align}

As stated, the values $\backslash sin(\backslash alpha)$ and $\backslash cos(\backslash alpha)$ appear to depend on the choice of a right triangle containing an angle of measure $\backslash alpha$. However, this is not the case as all such triangles are similar, and so the ratios are the same for each of them. For example, each leg of the 45-45-90 right triangle is 1 unit, and its hypotenuse is $\backslash sqrt\{2\}$; therefore, $\backslash sin\; 45^\backslash circ\; =\; \backslash cos\; 45^\backslash circ\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}$.{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA36 36]}} The following table shows the special value of each input for both sine and cosine with the domain between . The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be obtained by using a calculator.{{sfnp|Varberg|Purcell|Rigdon|2007|p=42}}{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA37 37], [https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA78 78]}}

{{Main|Law of sines|Law of cosines}}

File:Law of sines (simple).svg

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.{{sfnp|Axler|2012|p=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634 634]}} Given a triangle $ABC$ with sides $a$, $b$, and $c$, and angles opposite those sides $\backslash alpha$, $\backslash beta$, and $\backslash gamma$, the law states,

$$\backslash frac\{\backslash sin\; \backslash alpha\}\{a\}\; =\; \backslash frac\{\backslash sin\; \backslash beta\}\{b\}\; =\; \backslash frac\{\backslash sin\; \backslash gamma\}\{c\}.$$

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

$$\backslash frac\{a\}\{\backslash sin\; \backslash alpha\}\; =\; \backslash frac\{b\}\{\backslash sin\; \backslash beta\}\; =\; \backslash frac\{c\}\{\backslash sin\; \backslash gamma\}\; =\; 2R,$$

where $R$ is the triangle's circumradius.

The law of cosines is useful for computing the length of an unknown side if two other sides and an angle are known.{{sfnp|Axler|2012|p=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA634 634]}} The law states,

$$a^2\; +\; b^2\; -\; 2ab\backslash cos(\backslash gamma)\; =\; c^2$$

In the case where $\backslash gamma\; =\; \backslash pi/2$ from which $\backslash cos(\backslash gamma)\; =\; 0$, the resulting equation becomes the Pythagorean theorem.{{sfnp|Axler|2012|p=[https://books.google.com/books?id=B5RxDwAAQBAJ&pg=PA632 632]}}

The cross product and dot product are operations on two vectors in Euclidean vector space. The sine and cosine functions can be defined in terms of the cross product and dot product. If $\backslash mathbb\{a\}$ and $\backslash mathbb\{b\}$ are vectors, and $\backslash theta$ is the angle between $\backslash mathbb\{a\}$ and $\backslash mathbb\{b\}$, then sine and cosine can be defined as:

$$\backslash begin\{align\}$$

\sin (\theta) &= \frac

\cos (\theta) &= \frac{\mathbb{a} \cdot \mathbb{b}}

\end{align}

The sine and cosine functions may also be defined in a more general way by using unit circle, a circle of radius one centered at the origin $(0,0)$, formulated as the equation of $x^2\; +\; y^2\; =\; 1$ in the Cartesian coordinate system. Let a line through the origin intersect the unit circle, making an angle of $\backslash theta$ with the positive half of the {{nowrap|1=$x$-}}axis. The {{nowrap|1=$x$-}} and {{nowrap|1=$y$-}}coordinates of this point of intersection are equal to $\backslash cos\; (\backslash theta)$ and $\backslash sin\; (\backslash theta)$, respectively; that is,{{sfnp|Varberg|Purcell|Rigdon|2007|p=41}}

$$\backslash sin\; (\backslash theta)\; =\; y,\; \backslash qquad\; \backslash cos\; (\backslash theta)\; =\; x.$$

This definition is consistent with the right-angled triangle definition of sine and cosine when because the length of the hypotenuse of the unit circle is always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the {{nowrap|1=$y$-}}coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when , even under the new definition using the unit circle.{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA68 68]}}{{sfnp|Varberg|Purcell|Rigdon|2007|p=47}}

File:Circle cos sin.gif (in green), at an angle of {{math|1=θ}}. The cosine (in blue) is the {{nowrap|1={{math|1=x}}-}}coordinate.]]

Using the unit circle definition has the advantage of drawing a graph of sine and cosine functions. This can be done by rotating counterclockwise a point along the circumference of a circle, depending on the input $\backslash theta\; >\; 0$. In a sine function, if the input is $\backslash theta\; =\; \backslash frac\{\backslash pi\}\{2\}$, the point is rotated counterclockwise and stopped exactly on the {{nowrap|1=$y$-}}axis. If $\backslash theta\; =\; \backslash pi$, the point is at the circle's halfway. If $\backslash theta\; =\; 2\backslash pi$, the point returned to its origin. This results that both sine and cosine functions have the range between $-1\; \backslash le\; y\; \backslash le\; 1$.{{sfnp|Varberg|Purcell|Rigdon|2007|p=41–42}}

Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for the cosine function as well, although the point is rotated initially from the {{nowrap|1=$y$-}}coordinate. In other words, both sine and cosine functions are periodic, meaning any angle added by the circumference's circle is the angle itself. Mathematically,{{sfnp|Varberg|Purcell|Rigdon|2007|p=41, 43}}

$$\backslash sin(\backslash theta\; +\; 2\backslash pi)\; =\; \backslash sin(\backslash theta),\; \backslash qquad\; \backslash cos(\backslash theta\; +\; 2\backslash pi)\; =\; \backslash cos\; (\backslash theta).$$

A function $f$ is said to be odd if $f(-x)\; =\; -f(x)$, and is said to be even if $f(-x)\; =\; f(x)$. The sine function is odd, whereas the cosine function is even.{{sfnp|Young|2012|p=[https://books.google.com/books?id=OMrcN0a3LxIC&pg=RA1-PA165 165]}} Both sine and cosine functions are similar, with their difference being shifted by $\backslash frac\{\backslash pi\}\{2\}$. This means,{{sfnp|Varberg|Purcell|Rigdon|2007|p=42, 47}}

$$\backslash begin\{align\}$$

\sin(\theta) &= \cos\left(\frac{\pi}{2} - \theta \right), \\

\cos(\theta) &= \sin\left(\frac{\pi}{2} - \theta \right).

\end{align}

File:Cosine_fixed_point.svg

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 $\backslash sin(0)=0$. The only real fixed point of the cosine function is called the Dottie number. The Dottie number is the unique real root of the equation $\backslash cos\; (x)\; =\; x$. The decimal expansion of the Dottie number is approximately 0.739085.{{Cite web|url=https://oeis.org/A003957|title=OEIS A003957|website=oeis.org|access-date=2019-05-26}}

{{main|Differentiation of trigonometric functions}}

File:Sine quads 01 Pengo.svg]]

The sine and cosine functions are infinitely differentiable.{{sfnp|Bourchtein|Bourchtein|2022|p=[https://books.google.com/books?id=nGxOEAAAQBAJ&pg=PA294 294]}} The derivative of sine is cosine, and the derivative of cosine is negative sine:{{sfnp|Varberg|Purcell|Rigdon|2007|p=115}}

$$\backslash frac\{d\}\{dx\}\backslash sin(x)\; =\; \backslash cos(x),\; \backslash qquad\; \backslash frac\{d\}\{dx\}\backslash cos(x)\; =\; -\backslash sin(x).$$

Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself.{{sfnp|Bourchtein|Bourchtein|2022|p=[https://books.google.com/books?id=nGxOEAAAQBAJ&pg=PA294 294]}} These derivatives can be applied to the first derivative test, according to which the monotonicity of a function can be defined as the inequality of function's first derivative greater or less than equal to zero.{{sfnp|Varberg|Purcell|Rigdon|2007|p=155}} It can also be applied to second derivative test, according to which the concavity of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero.{{sfnp|Varberg|Purcell|Rigdon|2007|p=157}} The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign ($+$) denotes a graph is increasing (going upward) and the negative sign ($-$) is decreasing (going downward)—in certain intervals.{{sfnp|Varberg|Rigdon|Purcell|2007|p=42}} This information can be represented as a Cartesian coordinates system divided into four quadrants.

Both sine and cosine functions can be defined by using differential equations. The pair of $(\backslash cos\; \backslash theta,\; \backslash sin\; \backslash theta)$ is the solution $(x(\backslash theta),\; y(\backslash theta))$ to the two-dimensional system of differential equations $y\text{'}(\backslash theta)\; =\; x(\backslash theta)$ and $x\text{'}(\backslash theta)\; =\; -y(\backslash theta)$ with the initial conditions $y(0)\; =\; 0$ and $x(0)\; =\; 1$. 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. It can be interpreted as a phase space trajectory of the system of differential equations $y\text{'}(\backslash theta)\; =\; x(\backslash theta)$ and $x\text{'}(\backslash theta)\; =\; -y(\backslash theta)$ starting from the initial conditions $y(0)\; =\; 0$ and $x(0)\; =\; 1$.{{citation needed|date=August 2024}}

{{main|List of integrals of trigonometric functions}}

Their area under a curve can be obtained by using the integral with a certain bounded interval. Their antiderivatives are:

$$\backslash int\; \backslash sin(x)\backslash ,dx\; =\; -\backslash cos(x)\; +\; C\; \backslash qquad\; \backslash int\; \backslash cos(x)\backslash ,dx\; =\; \backslash sin(x)\; +\; C,$$

where $C$ denotes the constant of integration.{{sfnp|Varberg|Purcell|Rigdon|2007|p=199}} These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the arc length of the sine curve between $0$ and $t$ is

$$\backslash int\_0^t\backslash !\backslash sqrt\{1+\backslash cos^2(x)\}\backslash ,\; dx\; =\backslash sqrt\{2\}\; \backslash operatorname\{E\}\; \backslash left(t,\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; \backslash right),$$

where $\backslash operatorname\{E\}(\backslash varphi,k)$ is the incomplete elliptic integral of the second kind with modulus $k$. It cannot be expressed using elementary functions.{{sfnp|Vince|2023|p=[https://books.google.com/books?id=GnW6EAAAQBAJ&pg=PA162 162]}} In the case of a full period, its arc length is

$$L\; =\; \backslash frac\{4\backslash sqrt\{2\backslash pi\; ^3\}\}\{\backslash Gamma(1/4)^2\}\; +\; \backslash frac\{\backslash Gamma(1/4)^2\}\{\backslash sqrt\{2\backslash pi\}\}\; =\; \backslash frac\{2\backslash pi\}\{\backslash varpi\}+2\backslash varpi\; \backslash approx\; 7.6404\backslash ldots$$

where $\backslash Gamma$ is the gamma function and $\backslash varpi$ is the lemniscate constant.{{sfnp|Adlaj|2012}}

File:Arcsine_Arccosine.svg

The inverse function of sine is arcsine or inverse sine, denoted as "arcsin", "asin", or $\backslash sin^\{-1\}$.{{sfnp|Varberg|Purcell|Rigdon|2007|p=366}} The inverse function of cosine is arccosine, denoted as "arccos", "acos", or $\backslash cos^\{-1\}$.{{efn|1=The superscript of −1 in $\backslash sin^\{-1\}$ and $\backslash cos^\{-1\}$ denotes the inverse of a function, instead of exponentiation.}} As sine and cosine are not injective, their inverses are not exact inverse functions, but partial inverse functions. For example, $\backslash sin\; (0)\; =\; 0$, but also $\backslash sin\; (\backslash pi)\; =\; 0$, $\backslash sin\; (2\backslash pi)\; =\; 0$, and so on. It follows that the arcsine function is multivalued: $\backslash arcsin\; (0)\; =\; 0$, but also $\backslash arcsin\; (0)\; =\; \backslash pi$, $\backslash arcsin\; (0)\; =\; 2\backslash pi$, and so on. 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 $\backslash arcsin(x)$ will evaluate only to a single value, called its principal value. The standard range of principal values for arcsin is from , and the standard range for arccos is from $0$ to $\backslash pi$.{{sfnp|Varberg|Purcell|Rigdon|2007|p=365}}

The inverse function of both sine and cosine are defined as:{{citation needed|date=August 2024}}

$$\backslash theta\; =\; \backslash arcsin\; \backslash left(\; \backslash frac\{\backslash text\{opposite\}\}\{\backslash text\{hypotenuse\}\}\; \backslash right)\; =\; \backslash arccos\; \backslash left(\; \backslash frac\{\backslash text\{adjacent\}\}\{\backslash text\{hypotenuse\}\}\; \backslash right),$$

where for some integer $k$,

$$\backslash begin\{align\}$$

\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{align}

By definition, both functions satisfy the equations:{{citation needed|date=August 2024}}

$$\backslash sin(\backslash arcsin(x))\; =\; x\; \backslash qquad\; \backslash cos(\backslash arccos(x))\; =\; x$$

and

\arccos(\cos(\theta)) = \theta\quad & \text{for}\quad 0 \leq \theta \leq \pi\end{align}

{{Main|List of trigonometric identities}}

According to Pythagorean theorem, the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the Pythagorean trigonometric identity, the sum of a squared sine and a squared cosine equals 1:{{sfnp|Young|2017|p=[https://books.google.com/books?id=476ZDwAAQBAJ&pg=PA99 99]}}{{efn|1=Here, $\backslash sin^2\; (x)$ means the squared sine function $\backslash sin\; (x)\; \backslash cdot\; \backslash sin\; (x)$.}}

$$\backslash sin^2\; (\backslash theta)\; +\; \backslash cos^2(\backslash theta)\; =\; 1.$$

Sine and cosine satisfy the following double-angle formulas:{{cite book |title=Precalculus with Calculus Previews |author1=Dennis G. Zill |edition= |publisher=Jones & Bartlett Publishers |year=2013 |isbn=978-1-4496-4515-1 |page=238 |url=https://books.google.com/books?id=dtS5M4lx7scC}} [https://books.google.com/books?id=dtS5M4lx7scC&pg=PA238 Extract of page 238]

$$\backslash begin\{align\}$$

\sin(2\theta) &= 2\sin(\theta)\cos(\theta), \\

\cos(2\theta) &= \cos^2(\theta) - \sin^2(\theta) \\

&= 2\cos^2(\theta) - 1 \\

&= 1 - 2\sin^2(\theta)

\end{align}

{{anchor|Sine squared function}}

File:SinSquared.png

The cosine double angle formula implies that sin² and cos² are, themselves, shifted and scaled sine waves. Specifically,{{cite web |title=Sine-squared function |url=https://calculus.subwiki.org/wiki/Sine-squared_function#Identities |access-date=August 9, 2019}}

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

The graph shows both sine and sine squared functions, with the sine in blue and the 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.{{citation needed|date=August 2024}}

File:Sine.gif

Both sine and cosine functions can be defined by using a Taylor series, a power series involving the higher-order derivatives. As mentioned in {{slink||Continuity and differentiation}}, the derivative of sine is cosine and that the derivative of cosine is the negative of sine. This means the successive derivatives of $\backslash sin(x)$ are $\backslash cos(x)$, $-\backslash sin(x)$, $-\backslash cos(x)$, $\backslash sin(x)$, continuing to repeat those four functions. The {{nowrap|1=$(4n\; +\; k)$-}}th derivative, evaluated at the point 0:

$$\backslash sin^\{(4n+k)\}(0)=\backslash begin\{cases\}$$

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.{{sfnp|Varberg|Purcell|Rigdon|2007|p=491–492}} More generally, for all complex numbers:{{sfnp|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA74 74]}}

$$\backslash begin\{align\}$$

\sin(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\

&= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}

\end{align}

Taking the derivative of each term gives the Taylor series for cosine:{{sfnp|Varberg|Purcell|Rigdon|2007|p=491–492}}{{sfnp|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA74 74]}}

$$\backslash begin\{align\}$$

\cos(x) &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\

&= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}

\end{align}

Both sine and cosine functions with multiple angles may appear as their linear combination, resulting in a polynomial. Such a polynomial is known as the trigonometric polynomial. The trigonometric polynomial's ample applications may be acquired in its interpolation, and its extension of a periodic function known as the Fourier series. Let $a\_n$ and $b\_n$ be any coefficients, then the trigonometric polynomial of a degree $N$—denoted as $T(x)$—is defined as:{{sfnp|Powell|1981|p=150}}{{sfnp|Rudin|1987|p=88 }}

$$T(x)\; =\; a\_0\; +\; \backslash sum\_\{n=1\}^N\; a\_n\; \backslash cos\; (nx)\; +\; \backslash sum\_\{n=1\}^N\; b\_n\; \backslash sin\; (nx).$$

The trigonometric series can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let $A\_n$ and $B\_n$ be any coefficients, then the trigonometric series can be defined as:{{sfnp|Zygmund|1968|p=1}}

$$\backslash frac\{1\}\{2\}\; A\_0\; +\; \backslash sum\_\{n=1\}^\backslash infty\; A\_n\; \backslash cos\; (nx)\; +\; B\_n\; \backslash sin\; (nx).$$

In the case of a Fourier series with a given integrable function $f$, the coefficients of a trigonometric series are:{{sfnp|Zygmund|1968|p=11}}

$$\backslash begin\{align\}$$

A_n &= \frac{1}{\pi} \int_0 ^{2\pi} f(x) \cos (nx) \, dx, \\

B_n &= \frac{1}{\pi} \int_0 ^{2\pi} f(x) \sin (nx) \, dx.

\end{align}

{{more citations needed section|date=August 2024}}

Both sine and cosine can be extended further via complex number, a set of numbers composed of both real and imaginary numbers. For real number $\backslash theta$, the definition of both sine and cosine functions can be extended in a complex plane in terms of an exponential function as follows:{{sfnp|Howie|2003|p=[https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24 24]}}

$$\backslash begin\{align\}$$

\sin (\theta) &= \frac{e^{i \theta} - e^{-i \theta}}{2i}, \\

\cos (\theta) &= \frac{e^{i \theta} + e^{-i \theta}}{2},

\end{align}

Alternatively, both functions can be defined in terms of Euler's formula:{{sfnp|Howie|2003|p=[https://books.google.com/books?id=0FZDBAAAQBAJ&pg=PA24 24]}}

$$\backslash begin\{align\}$$

e^{i \theta} &= \cos (\theta) + i \sin (\theta), \\

e^{-i \theta} &= \cos (\theta) - i \sin (\theta).

\end{align}

When plotted on the complex plane, the function $e^\{ix\}$ for real values of $x$ traces out the unit circle in the complex plane. Both sine and cosine functions may be simplified to the imaginary and real parts of $e^\{i\; \backslash theta\}$ as:{{sfnp|Rudin|1987|p=2}}

$$\backslash begin\{align\}$$

\sin \theta &= \operatorname{Im}(e^{i \theta}), \\

\cos \theta &= \operatorname{Re}(e^{i \theta}).

\end{align}

When $z=x+iy$ for real values $x$ and $y$, where $i\; =\; \backslash sqrt\{-1\}$, both sine and cosine functions can be expressed in terms of real sines, cosines, and hyperbolic functions as:{{citation needed|date=August 2024}}

$$\backslash begin\{align\}$$

\sin z &= \sin x \cosh y + i \cos x \sinh y, \\

\cos z &= \cos x \cosh y - i \sin x \sinh y.

\end{align}

File:Sinus und Kosinus am Einheitskreis 3.svg

Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates $(r,\; \backslash theta)$:

$$z\; =\; r(\backslash cos(\backslash theta)\; +\; i\backslash sin(\backslash theta)),$$

and the real and imaginary parts are

$$\backslash begin\{align\}$$

\operatorname{Re}(z) &= r \cos(\theta), \\

\operatorname{Im}(z) &= r \sin(\theta),

\end{align}

where $r$ and $\backslash theta$ represent the magnitude and angle of the complex number $z$.

For any real number $\backslash theta$, Euler's formula in terms of polar coordinates is stated as $z\; =\; re^\{i\backslash theta\}$.

File:Complex_sin.jpg of sin(z) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.]]

File:Sin z vector field 02 Pengo.svg

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

: $\backslash begin\{align\}$

\sin(z)& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}z^{2n+1} \\

& = \frac{e^{i z} - e^{-i z}}{2i} \\

& = \frac{\sinh \left( i z\right) }{i} \\

& = -i \sinh \left(i z\right)\\

\cos(z)& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}z^{2n} \\

& = \frac{e^{i z} + e^{-i z}}{2} \\

& = \cosh( i z) \\

\end{align}

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:

: $\backslash begin\{align\}$

\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{align}

Using the partial fraction expansion technique in complex analysis, one can find that the infinite series

$$\backslash sum\_\{n\; =\; -\backslash infty\}^\{\backslash infty\}\backslash frac\{(-1)^n\}\{z-n\}\; =\; \backslash frac\{1\}\{z\}\; -2z\; \backslash sum\_\{n\; =\; 1\}^\{\backslash infty\}\backslash frac\{(-1)^n\}\{n^2\; -\; z^2\}$$

both converge and are equal to $\backslash frac\{\backslash pi\}\{\backslash sin\; (\backslash pi\; z)\}$. Similarly, one can show that

$$\backslash frac\{\backslash pi^2\}\{\backslash sin^2\; (\backslash pi\; z)\}\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; \backslash frac\{1\}\{(z\; -\; n)^2\}.$$

Using product expansion technique, one can derive

$$\backslash sin(\backslash pi\; z)\; =\; \backslash pi\; z\; \backslash prod\_\{n\; =\; 1\}^\backslash infty\; \backslash left(\; 1\; -\; \backslash frac\{z^2\}\{n^2\}\; \backslash right).$$

sin(z) is found in the functional equation for the Gamma function,

:$\backslash Gamma(s)\backslash Gamma(1\; -\; s)\; =\; \{\backslash pi\backslash over\backslash sin(\backslash pi\; s)\},$

which in turn is found in the functional equation for the Riemann zeta-function,

:$\backslash zeta(s)\; =\; 2(2\backslash pi)^\{s-1\}\backslash Gamma(1\; -\; s)\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{2\}\; s\backslash right)\backslash zeta(1\; -\; s).$

As a holomorphic function, sin z is a 2D solution of Laplace's equation:

:$\backslash Delta\; u(x\_1,\; x\_2)\; =\; 0.$

The complex sine function is also related to the level curves of pendulums.{{how|reason=This does not explain how sine is related to pendulums.|date=August 2019}}{{cite web|url=https://math.stackexchange.com/q/220418|title=Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?|website=math.stackexchange.com|access-date=2019-08-12}}{{better source needed|date=August 2019}}

{{main|History of trigonometry#Etymology}}

The word sine is derived, indirectly, from the Sanskrit word {{lang|sa|jyā}} 'bow-string' or more specifically its synonym {{lang|sa|jīvá}} (both adopted from Ancient Greek {{lang|grc|χορδή}} 'string; chord'), 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; sine and chord are closely related in a circle of unit diameter, see Ptolemy’s Theorem). This was transliterated in Arabic as {{tlit|ar|jība}}, which is meaningless in that language and written as {{tlit|ar|jb}} ({{lang|ar|جب}}). Since Arabic is written without short vowels, {{tlit|ar|jb}} was interpreted as the homograph {{tlit|ar|jayb}} (جيب), which means 'bosom', 'pocket', or 'fold'.{{sfnp|Plofker|2009|p=[https://books.google.com/books?id=DHvThPNp9yMC&pg=PA257 257]}}{{sfnp|Maor|1998|p=[https://books.google.com/books?id=r9aMrneWFpUC&pg=PA35 35]}} 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').{{sfnp|Merzbach|Boyer|2011}}{{sfnp|Maor|1998|p=35–36}}{{sfnp|Katz|2008|p=253}} 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.{{sfnp|Smith|1958|p=202}}Various sources credit the first use of {{Lang|la-x-medieval|sinus}} to either

• Plato Tiburtinus's 1116 translation of the Astronomy of Al-Battani
• Gerard of Cremona's translation of the Algebra of al-Khwārizmī
• Robert of Chester's 1145 translation of the tables of al-Khwārizmī

See {{harvp|Merlet|2004}}. See {{harvp|Maor|1998}}, Chapter 3, for an earlier etymology crediting Gerard. See {{harvp|Katz|2008|p=210}}. The English form sine was introduced in Thomas Fale's 1593 Horologiographia.Fale's book alternately uses the spellings "sine", "signe", or "sign". {{pb}} {{cite book |last=Fale |first=Thomas |title=Horologiographia. The Art of Dialling: Teaching, an Easie and Perfect Way to make all Kindes of Dials .... |page=11, for example |url=https://archive.org/details/b30333106/page/19/mode/1up |year=1593 |place=London |publisher=F. Kingston }}

The word cosine derives from an abbreviation of the Latin {{lang|la|complementi sinus}} 'sine of the complementary angle' as cosinus in Edmund Gunter's Canon triangulorum (1620), which also includes a similar definition of cotangens.{{sfnp|Gunter|1620}}

{{Main|History of trigonometry}}

File:Khalili Collection Islamic Art sci 0040.1 CROP.jpg with axes for looking up the sine and versine of angles]]

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).{{Cite journal |last=Brendan |first=T. |date=February 1965 |title=How Ptolemy constructed trigonometry tables |journal=The Mathematics Teacher |volume=58 |issue=2 |pages=141–149 |doi=10.5951/MT.58.2.0141 |jstor=27967990 }}

The sine and cosine functions are closely related to the Jyā, koti-jyā and utkrama-jyā functions used in Indian astronomy during the Gupta period (Aryabhatiya and Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.{{sfnp|Merzbach|Boyer|2011}}{{cite book |last=Van Brummelen |first=Glen |author-link=Glen Van Brummelen |year=2009 |title=The Mathematics of the Heavens and the Earth |chapter=India |at=Ch. 3, {{pgs|94–134}} |publisher=Princeton University Press |isbn=978-0-691-12973-0}}

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.{{cite magazine |title=Islamic Astronomy |author-first=Owen |author-last=Gingerich |magazine=Scientific American |date=1986 |volume=254 |page=74 |url=http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |access-date=2010-07-13 |archive-url=https://web.archive.org/web/20131019140821/http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |archive-date=2013-10-19}} Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents.Jacques Sesiano, "Islamic mathematics", p. 157, in {{Cite book |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=Springer Science+Business Media |isbn=978-1-4020-0260-1}}{{cite web |title=trigonometry |date=17 June 2024 |url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |publisher=Encyclopedia Britannica}} Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.

In the early 17th-century, the French mathematician Albert Girard published the first use of the abbreviations sin, cos, and tan; these were further promulgated by Euler (see below). The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.

In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.{{cite book|title=Elements of the History of Mathematics|url=https://archive.org/details/elementsofhistor0000bour|url-access=registration|author=Nicolás Bourbaki|publisher=Springer|year=1994|isbn=9783540647676}} Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722)."[http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf Why the sine has a simple derivative] {{webarchive|url=https://web.archive.org/web/20110720102700/http://www.math.usma.edu/people/rickey/hm/CalcNotes/Sine-Deriv.pdf |date=2011-07-20 }}", in [http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm Historical Notes for Calculus Teachers] {{webarchive|url=https://web.archive.org/web/20110720102613/http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm |date=2011-07-20 }} by [http://www.math.usma.edu/people/rickey/ V. Frederick Rickey] {{webarchive|url=https://web.archive.org/web/20110720102654/http://www.math.usma.edu/people/rickey/ |date=2011-07-20 }} Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.{{sfnp|Merzbach|Boyer|2011}}

{{more citations needed section|date=August 2024}}

{{See also|Lookup table#Computing sines}}

There is no standard algorithm for calculating sine and cosine. IEEE 754, the most widely used standard for the specification of reliable floating-point computation, does not address calculating trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.{{sfnp|Zimmermann|2006}}

Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. sin(10{{sup|22}}).

A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest pre-calculated value, or linearly interpolate between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.{{citation needed|date=October 2012}}

The CORDIC algorithm is commonly used in scientific calculators.

The sine and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to sin and cos.

Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.

In programming languages, sin and cos are typically either a built-in function or found within the language's standard math library. For example, the C standard library defines sine functions within math.h: sin(double), sinf(float), and sinl(long double). The parameter of each is a floating point value, specifying the angle in radians. Each function returns the same data type as it accepts. Many other trigonometric functions are also defined in math.h, such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly, Python defines math.sin(x) and math.cos(x) within the built-in math module. Complex sine and cosine functions are also available within the cmath module, e.g. cmath.sin(z). CPython's math functions call the C math library, and use a double-precision floating-point format.

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 $\backslash pi$ radians. Representing angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases."[https://www.mathworks.com/help/matlab/ref/double.sinpi.html MATLAB Documentation sinpi]"[https://www.rdocumentation.org/packages/base/versions/3.5.3/topics/Trig R Documentation sinpi] These functions are called sinpi and cospi in MATLAB, OpenCL,"[https://www.khronos.org/registry/OpenCL/sdk/1.0/docs/man/xhtml/sin.html OpenCL Documentation sinpi] R, Julia,"[http://www.jlhub.com/julia/manual/en/function/sinpi Julia Documentation sinpi] CUDA,"[https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__DOUBLE.html CUDA Documentation sinpi] and ARM."[https://developer.arm.com/docs/100614/latest/b-opencl-built-in-functions/b2-math-functions ARM Documentation sinpi] For example, sinpi(x) would evaluate to $\backslash sin(\backslash pi\; x),$ where x is expressed in half-turns, and consequently the final input to the function, {{math|πx}} can be interpreted in radians by {{math|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 $2\backslash pi$, $\backslash pi$, and $\backslash frac\{\backslash pi\}\{2\}$ 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 $\backslash frac\{\backslash pi\}\{2\}$ involves inaccuracies in representing $\backslash frac\{\backslash pi\}\{2\}$.

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."[https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx ALLEGRO Angle Sensor Datasheet] {{Webarchive|url=https://web.archive.org/web/20190417143715/https://www.allegromicro.com/en/Products/Magnetic-Linear-And-Angular-Position-Sensor-ICs/Angular-Position-Sensor-ICs/AAS33051.aspx |date=2019-04-17 }} 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 $\backslash frac\{\backslash pi\}\{2048\}$ would be incurred.

{{div col|colwidth=20em}}

• Āryabhaṭa's sine table
• Bhaskara I's sine approximation formula
• Discrete sine transform
• Dixon elliptic functions
• Euler's formula
• Generalized trigonometry
• Hyperbolic function
• Lemniscate elliptic functions
• Law of sines
• List of periodic functions
• List of trigonometric identities
• Madhava series
• Madhava's sine table
• Optical sine theorem
• Polar sine—a generalization to vertex angles
• Proofs of trigonometric identities
• Sinc function
• Sine and cosine transforms
• Sine integral
• Sine quadrant
• Sine wave
• Sine–Gordon equation
• Sinusoidal model
• SOH-CAH-TOA
• Trigonometric functions
• Trigonometric integral

{{div col end}}

{{notelist|group="efn"}}

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{{refend}}

{{Wiktionary|sine}}

• {{Commons category-inline|Sine function}}

{{Wiktionary}}

{{Trigonometric and hyperbolic functions}}

Category:Angle

Category:Trigonometric functions

no:Trigonometriske funksjoner#Sinus, cosinus og tangens