Sine and cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle, the sine and cosine functions are denoted as and.
The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine and cosine functions are 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. They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period.
Elementary descriptions
Right-angled triangle definition
To define the sine and cosine of an acute angle, start with a right triangle that contains an angle of measure ; in the accompanying figure, angle in a right triangle 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, it is.
- The hypotenuse is the side opposite the right angle; in this case, it is. The hypotenuse is always the longest side of a right-angled triangle.
- The adjacent side is the remaining side; in this case, it is. It forms a side of both the angle of interest and the right angle.
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:
Special angle measures
As stated, the values and appear to depend on the choice of a right triangle containing an angle of measure. 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 ; therefore,. 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.Laws
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. Given a triangle with sides,, and, and angles opposite those sides,, and, the law states,This is equivalent to the equality of the first three expressions below:
where 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. The law states,
In the case where from which, the resulting equation becomes the Pythagorean theorem.
Vector definition
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 and are vectors, and is the angle between and, then sine and cosine can be defined as:Analytic descriptions
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, formulated as the equation of in the Cartesian coordinate system. Let a line through the origin intersect the unit circle, making an angle of with the positive half of the axis. The and coordinates of this point of intersection are equal to and, respectively; that is,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 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.
Graph of a function and its elementary properties
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. In a sine function, if the input is, the point is rotated counterclockwise and stopped exactly on the axis. If, the point is at the circle's halfway point. If, the point returns to its origin. This results in both sine and cosine functions having the range between.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,
A function is said to be odd if, and is said to be even if. The sine function is odd, whereas the cosine function is even. Both sine and cosine functions are similar, with their difference being shifted by. This phase shift can be expressed as or. This is distinct from the cofunction identities that follow below, which arise from right-triangle geometry and are not phase shifts:
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. 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. 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: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 and the negative sign is decreasing —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 is the solution to the two-dimensional system of differential equations and with the initial conditions and. 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 and starting from the initial conditions and.
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:where 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 and is
where is the incomplete elliptic integral of the second kind with modulus. It cannot be expressed using elementary functions. In the case of a full period, its arc length is
where is the gamma function and is the lemniscate constant.
Inverse functions
The inverse function of sine is arcsine or inverse sine, denoted as "arcsin", "asin", or. The inverse function of cosine is arccosine, denoted as "arccos", "acos", or. As sine and cosine are not injective, their inverses are not exact inverse functions, but partial inverse functions. For example,, but also, , and so on. It follows that the arcsine function is multivalued:, but also,, and so on. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each in the domain, the expression will evaluate only to a single value, called its principal value. The standard range of principal values for arcsin is from to, and the standard range for arccos is from to.The inverse function of both sine and cosine are defined as:
where for some integer,
By definition, both functions satisfy the equations:
and
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:Sine and cosine satisfy the following double-angle formulas:
The cosine double angle formula implies that sin2 and cos2 are, themselves, shifted and scaled sine waves. Specifically,
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.