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 (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle θ {\displaystyle \theta } , the sine and cosine functions are denoted as sin ( θ ) {\displaystyle \sin(\theta )} and cos ( θ ) {\displaystyle \cos(\theta )} . 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 α {\displaystyle \alpha } , start with a right triangle that contains an angle of measure α {\displaystyle \alpha } ; in the accompanying figure, angle α {\displaystyle \alpha } in a right triangle A B C {\displaystyle 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, it is a {\displaystyle a} . The hypotenuse is the side opposite the right angle; in this case, it is h {\displaystyle 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 {\displaystyle 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:
sin ( α ) = opposite hypotenuse , cos ( α ) = adjacent hypotenuse . {\displaystyle \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:
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