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These are the six basic trigonometric functions, together with their standard notational abbreviations. The last four functions are defined in terms of the first two. In other words, the four equations below are definitions, not proved identities.
Several relations between these functions are listed on the page about trigonometric identities.
A few other functions were common historically (and appeared in the earliest tables), but are now little-used, such as the versed sine (versin = 1 − cos) and the exsecant (exsec = sec − 1).
In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A:
We use the following names for the sides of the triangle:
Then,
1). The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case
Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.
2). The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case
3). The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case
The remaining three functions are best defined using the above three functions.
4). The cosecant csc(A) is the multiplicative inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:
5). The secant sec(A) is the multiplicative inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:
6). The cotangent cot(A) is the multiplicative inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:
There are a number of mnemonics for the above definitions, for example SOHCAHTOA (sounds like "soak a toe-a", can be read as "soccer tour"). It reminds one that:
Many other such words and phrases have been contrived; for more, see: trigonometry mnemonics.
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.)
Prior to computers, people typically evaluated trigonometric functions by interpolatingThis article is about interpolation in mathematics. See also interpolation (music . In the mathematical subfield of numerical analysis interpolation is a method of constructing new data points from a discrete set of known data points. According to the Oxf from a detailed table of their values, calculated to many significant figuresSignificant figures (also called sig figs significant digits or sig digs is a method of expressing errors in measurements. The term is also sometimes used to describe some rules-of-thumb, known as significance arithmetic which attempt to indicate the prop. 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 tablesTables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of stu.
Modern computers use a variety of techniques (Kantabutra, 1996). One common method, especially on higher-end processors with floating pointComputer arithmetic A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. In particular, it represents an integer or fixed-poi units, is to combine a polynomialIn mathematics polynomial functions or polynomials are an important class of simple and smooth functions. Simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i. they have derivatives o approximation (such as a Taylor seriesIn mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval a − r a + r is the power series : Here, n is the factorial of n and f n a denotes the n''th derivative of f at the point or a rational functionIn mathematics, a rational function is a ratio of polynomials. For a single variable x a typical rational function is therefore P ''x Q ''x where P and Q are polynomials in x as indeterminate, and Q isn't the zero polynomial. Any non-zero polynomial Q is) 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 multiplierALU redirects here. Alternative meaning: Alu sequence. An arithmetic and logical unit (ALU is one of the core components of all central processing units. It is capable of calculating the results of a wide variety of common computations. The most common avs, 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.
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 three degrees (π/60 radians) can be found exactly by hand.
Consider a right triangle where the two other angles are equal, and therefore are both 45 degrees (π/4 radians). 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 45 degrees can then be found using the Pythagorean theorem, c = √(a2 + b2) = √2.
Therefore,
To determine the trigonometric functions for angles of 60 degrees (π/3 radians) and 30 degrees (π/6 radians), we start with an equilateral triangle of side length 1. All its angles are 60 degrees. By dividing it into two, we obtain a right triangle with 30 and 60 degree angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields
and
see also: Exact trigonometric constants
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