Trigonometric function

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below.

Computation

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. In this section, however, we describe more details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found. (Below, it suffices to consider a small range of angles, say 0 to π/2, since all other angles can be reduced to this range by the periodicity and symmetries of the trigonometric functions.)

Related Topics:
Computer - Scientific calculator

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures. Such tables have been available for as long as trigonometric functions have been described (see History, below), and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2)=1). See also: Generating trigonometric tables.

Related Topics:
Interpolating - Significant figures - Identities - Generating trigonometric tables

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Modern computers use a variety of techniques (Kantabutra, 1996). One common method, especially on higher-end processors with floating point units, is to combine a polynomial approximation (such as a Taylor series or a rational function) with a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction. On simpler devices that lack hardware multipliers, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons.

Related Topics:
Floating point - Polynomial - Taylor series - Rational function - Hardware multiplier - CORDIC - Shift - Hardware

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in the following examples. In fact, the sine, cosine and tangent of any integer multiple of π/60 radians (three degrees) can be found exactly by hand.

Related Topics:
Pythagorean theorem - Exactly by hand

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Consider a right triangle where the two other angles are equal, and therefore are both π/4 radians (45 degrees). Then the length of side b and the length of side a are equal; we can choose a = b = 1. The values of sine, cosine and tangent of an angle of π/4 radians (45 degrees) can then be found using the Pythagorean theorem:

Related Topics:
Radian

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:c = sqrt { a^2+b^2 } = sqrt2

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Therefore:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sin left(pi / 4 ight) = sin left(45^circ ight) = cos left(pi / 4 ight) = cos left(45^circ ight) = {1 over sqrt2}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: an left(pi / 4 ight) = an left(45^circ ight) = {sqrt2 over sqrt2} = 1

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sin left(pi / 6 ight) = sin left(30^circ ight) = cos left(pi / 3 ight) = cos left(60^circ ight) = {1 over 2}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:cos left(pi / 6 ight) = cos left(30^circ ight) = sin left(pi / 3 ight) = sin left(60^circ ight) = {sqrt3 over 2}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: an left(pi / 6 ight) = an left(30^circ ight) = cot left(pi / 3 ight) = cot left(60^circ ight) = {1 over sqrt3}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

See also: Exact trigonometric constants

~ ~ ~ ~ ~ ~ ~ ~ ~ ~