Unit circle definition 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. A ray from the origin making an angle of \theta with the positive half of the axis intersects the unit circle at exactly one point. The and coordinates of this point of intersection are equal to \cos (\theta) and \sin (\theta) , respectively; that is, \sin (\theta) = y, \qquad \cos (\theta) = x. This definition is consistent with the right-angled triangle definition of sine and cosine when 0 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 coordinate. A similar argument can be made for the cosine function to show that the cosine of an angle when 0 , even under the new definition using the unit circle.
Graph of a function and its elementary properties (in green), at an angle of . The cosine (in blue) is the coordinate. Using the unit circle definition has the advantage of drawing the 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 \theta > 0 . In a sine function, if the input is \theta = \frac{\pi}{2} , the point is rotated counterclockwise and stopped exactly on the axis. If \theta = \pi , the point is at the circle's halfway point. If \theta = 2\pi , the point returns to its origin. This results in both sine and cosine functions having the
range between -1 \le y \le 1 . 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 coordinate. In other words, both sine and cosine functions are
periodic, meaning any angle added by the circle's circumference is the angle itself. Mathematically, \sin(\theta + 2\pi) = \sin(\theta), \qquad \cos(\theta + 2\pi) = \cos (\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. Both sine and cosine functions are similar, with their difference being
shifted by \frac{\pi}{2} . This phase shift can be expressed as \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) or \sin(\theta) = \cos\left(\theta - \frac{\pi}{2}\right) . This is distinct from the cofunction identities that follow below, which arise from right-triangle geometry and are not phase shifts: \begin{align} \sin(\theta) &= \cos\left(\frac{\pi}{2} - \theta \right), \\ \cos(\theta) &= \sin\left(\frac{\pi}{2} - \theta \right). \end{align} 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. 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 \cos (x) = x. The decimal expansion of the Dottie number is approximately 0.739085.
Continuity and differentiation The sine and cosine functions are infinitely differentiable. The derivative of sine is cosine, and the derivative of cosine is negative sine: \frac{d}{dx}\sin(x) = \cos(x), \qquad \frac{d}{dx}\cos(x) = -\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. 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. 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. 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. 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 (\cos \theta, \sin \theta) is the solution (x(\theta), y(\theta)) to the two-dimensional system of
differential equations y'(\theta) = x(\theta) and x'(\theta) = -y(\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'(\theta) = x(\theta) and x'(\theta) = -y(\theta) starting from the initial conditions y(0) = 0 and x(0) = 1.
Integral and the usage in mensuration Their area under a curve can be obtained by using the
integral with a certain bounded interval. Their antiderivatives are: \int \sin(x)\,dx = -\cos(x) + C \qquad \int \cos(x)\,dx = \sin(x) + C, where C denotes the
constant of integration. 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 \int_0^t\!\sqrt{1+\cos^2(x)}\, dx =\sqrt{2} \operatorname{E} \left(t, \frac{1}{\sqrt{2}} \right), where \operatorname{E}(\varphi,k) is the
incomplete elliptic integral of the second kind with modulus k. It cannot be expressed using
elementary functions. In the case of a full period, its arc length is L = \frac{4\sqrt{2\pi ^3}}{\Gamma(1/4)^2} + \frac{\Gamma(1/4)^2}{\sqrt{2\pi}} = \frac{2\pi}{\varpi}+2\varpi \approx 7.6404 where \Gamma is the
gamma function and \varpi is the
lemniscate constant.
Inverse functions The functions \sin : \mathbb{R}\to \mathbb{R} and \cos : \mathbb{R}\to \mathbb{R} (as well as those functions with the same function rule and domain whose codomain is a subset of \mathbb{R} containing the interval \left[ -1,1\right]) are not bijective and therefore do not have inverse functions. For example, \sin (0) = 0 , but also \sin (\pi) = 0 , \sin (2\pi) = 0 . Sine's "inverse", called arcsine, can then be described not as a function but a relation (for example, all integer multiples of \pi would have an arcsine of zero). To define the inverse functions of sine and cosine, they must be restricted to their
principal branches by restricting their domain and codomain; the standard functions used to define arcsine and arccosine are then \sin : \left[-\pi/2, \pi/2 \right]\to \left[-1, 1 \right] and \cos : \left[0,\pi\right]\to \left[-1,1 \right]. These are bijective and have inverses: \arcsin : \left[-1,1 \right]\to \left[-\pi/2,\pi/2 \right] and \arccos : \left[-1,1 \right]\to \left[ 0, \pi\right]. Alternative notation is \sin^{-1} for arcsine and \cos^{-1} for arccosine. Using these definitions, one obtains the identity maps: \begin{align} \sin\circ\arcsin\,(x)&=x\qquad x\in \left[ -1,1\right]\\ \arcsin\circ\sin\, (x)&=x\qquad x\in \left[ -\pi/2, \pi/2\right] \end{align}and \begin{align} \cos\circ\arccos\, (x)&=x\qquad x\in \left[-1,1\right]\\ \arccos\circ\cos\,(x)&=x\qquad x\in \left[0,\pi\right] \end{align} An acute angle \theta is given by: \theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right), where for some integer k , \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: \sin(\arcsin(x)) = x \qquad \cos(\arccos(x)) = x and \begin{align}\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{align}
Other 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: \sin^2 (\theta) + \cos^2(\theta) = 1. Sine and cosine satisfy the following double-angle formulas: \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} The cosine double angle formula implies that sin2 and cos2 are, themselves, shifted and scaled sine waves. Specifically, \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}\qquad\cos^2(\theta) = \frac{1 + \cos(2\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.
Series and polynomials Both sine and cosine functions can be defined by using a
Taylor series, a
power series involving the higher-order derivatives. As mentioned in , the
derivative of sine is cosine and 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 th derivative, evaluated 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} 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. More generally, for all
complex numbers: \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: \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: T(x) = a_0 + \sum_{n=1}^N a_n \cos (nx) + \sum_{n=1}^N b_n \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: \frac{1}{2} A_0 + \sum_{n=1}^\infty A_n \cos (nx) + B_n \sin (nx). In the case of a Fourier series with a given integrable function f , the coefficients of a trigonometric series are: \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} == Complex numbers relationship ==