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Trigonometric function

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In mathematics, the trigonometric functions are functions of an angle important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios of coordinates of points on the unit circle, or, more generally still, as infinite series. All three approaches will be presented below.

There are six basic trigonometric functions.

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Secant (sec) ( 1 / cos )
• Cosecant (csc) ( 1 / sin )
• Cotangent (cot) ( 1 / tan )

Sine, cosine and tangent are by far the most important. Several relations between these functions are listed on the page about trigonometric identities.

Right Triangle Definitions

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:

• The hypotenuse is the side opposite the right angle, in this case c.
• The opposite side is the side opposite to the angle we are interested in, in this case a.
• The adjacent side is the side that is a leg of the angle, but not the hypotenuse, in this case b.

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

sin(A) = opp/hyp = a/c.
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.

A mnemonic commonly used in the UK is "OHMS". This is memorable because it might mean "On Her Majesty's Service", which is stamped on the front of mail sent by the government, or "Opposite over Hypotenuse Means Sine".

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

One familiar mnemonic to remember these definitions is SOHCAHTOA. It reminds one that "SOH", sin = opposite/hypotenuse,"CAH", cos = adjacent/hypotenuse, and "TOA", tan = opposite/adjacent.

The remaining three functions are best defined using the above three functions.

4). The cosecant csc(A) is the inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:

csc(A) = hyp/opp = c/a.

5). The secant sec(A) is the inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:

6). The cotangent cot(A) is the inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:

Computing

The values of the trigonometric functions have been tabulated and can also be computed by calculator. For some simple angles, the values can be computed by hand, as in the following examples:

Suppose we have a right triangle where the two other angles are equal, and therefore = 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. Now, one can determine the sin, cos and tan of an angle of 45 degrees. Using the Pythagorean Theorem, c = √(a2 + b2) = √2. This is illustrated in the following figure:

Therefore,

$\sin \left(45^\circ\right) = {1 / \sqrt2} = {\sqrt2 / 2}$
$\cos \left(45^\circ\right) = {1 / \sqrt2} = {\sqrt2 / 2}$
$\tan \left(45^\circ\right) = {\sqrt2 / \sqrt2} = 1$

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

$\sin \left(30^\circ\right) = {1 / 2}$
$\cos \left(30^\circ\right) = {\sqrt3 / 2}$
$\tan \left(30^\circ\right) = {\sqrt3 / 3}$

and

$\sin \left(60^\circ\right) = {\sqrt3 / 2}$
$\cos \left(60^\circ\right) = {1 / 2}$
$\tan \left(60^\circ\right) = \sqrt3$

Unit Circle Definitions

The six trig functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. The unit circle definition does, however, permit the definition of the trig functions for all positive and negative arguments, not just for angles between 0 and π/2 radians.

The equation for the unit circle is:

$x^2 + y^2 = 1$

and it looks like this:

In the picture, some common angles, measured in radians, are given. Note that we measure angles positive in the counter clockwise direction and angles negative in the clockwise direction. The coordinates of where a line that makes an angle θ with the positive half of the x-axis intersects the circle are equal to cosθ and sinθ, respectively. The triangle in the graphic reveals the reason: the radius is equal to the hypotenuse and has length 1, sinθ = y/1 and cosθ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the length of their hypotenuses equal to 1.

For angles greater than 2π or less than -2π simply continue to rotate around the circle. In this way, sine and cosine become periodic functions with period 2π:

$\sin\theta = \sin\left(\theta + 2k\pi\right)$
$\cos\theta = \cos\left(\theta + 2k\pi\right)$

for any angle θ and any integer k.

Though only sine and cosine were defined directly by the unit circle, the other four trig functions can be defined by

$\tan\theta = \frac{\sin\theta}{\cos\theta}$
$\sec\theta = \frac{1}{\cos\theta}$
$\csc\theta = \frac{1}{\sin\theta}$
$\cot\theta = \frac{\cos\theta}{\sin\theta}$

Here is a plot of sine and cosine:

Series definitions

Here, and generally in calculus, it is of utmost importance that all angles are measured in radians. One may then define

$\sin\left(x\right) = x - {x^3 / 3!} + {x^5 / 5!} - {x^7 / 7!} + \cdots$
$\cos\left(x\right) = 1 - {x^2 / 2!} + {x^4 / 4!} - {x^6 / 6!} + \cdots$

These definitions are equivalent to the above given ones because of the theory of Taylor series, and because of the fact that the derivative of sine is cosine and the derivative of cosine is -sine. These definitions are often used as the starting point in a rigorous treatment of analysis since the theory of such infinite series is well known. The differentiability and continuity is then easily established, as is Euler's formula relating the trigonometric functions to the exponential function as well as the most remarkable formula in the world. The series definitions have the additional advantage that they allow to extend the sine and cosine functions for all complex arguments.

Inverse Functions

The trigonometric functions are not monotonic, so their inverses are not unique. The principle inverses are usually defined as:

$y = \arcsin\left(x\right) \mbox{is equivalent to}\, x = \sin\left(y\right) \mbox{if}\, -{\pi / 2} \le y \le {\pi / 2}$
$y = \arccos\left(x\right) \mbox{is equivalent to}\, x = \cos\left(y\right) \mbox{if}\, 0 \le y \le \pi$
$y = \arctan\left(x\right) \mbox{is equivalent to}\, x = \tan\left(y\right) \mbox{if}\, -{\pi / 2} \le y \le {\pi / 2}$
$y = \arccsc\left(x\right) \mbox{is equivalent to}\, x = \csc\left(y\right) \mbox{if}\, -{\pi / 2} \le y \le {\pi / 2}$
$y = \arcsec\left(x\right) \mbox{is equivalent to}\, x = \sec\left(y\right) \mbox{if}\, 0 \le y \le \pi$
$y = \arccot\left(x\right) \mbox{is equivalent to}\, x = \cot\left(y\right) \mbox{if}\, 0 \le y \le \pi$

These functions are each equivalent to an integral:

$\arcsin\left(x\right) = \int \left(1 - x^2\right)^{-.5}dx$
$\arccos\left(x\right) = \int -\left(1 - x^2\right)^{-.5}dx$
$\arctan\left(x\right) = \int \left(1 + x^2\right)^{-1}dx$
$\arccsc\left(x\right) = \int \left(-x \left(x^2 - 1\right)^{.5}\right)^{-1}dx$
$\,\arcsec\left(x\right) = \int \left(x \left(x^2 - 1\right)^{.5}\right)^{-1}dx$
$\arccot\left(x\right) = \int -\left(x^2 + 1\right)^{-1}dx$

Note: arcsec also means arcsecond.

Properties and applications

The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results:

The law of sines for an arbitrary triangle states:

sin(A)/a = sin(B)/b = sin(C)/c
It can be proven by dividing the triangle into two right ones and using the above definition of sine. The common number sin(A)/a occurring in the theorem is the reciprocal of the diameter of the circle through the three points A, B and C. 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. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

If the angle is not contained between the two sides, the triangle may not be unique. Be aware of this ambiguous case of the Sine Law.

The law of cosines is an extension to the Pythagorean Theorem:

c2 = a2 + b2 - 2ab cos(C)
Again, this theorem can be proven by dividing the triangle into two right ones. The law of cosines is useful to determine the unknown data of a triangle if two sides and an angle are known.

There is also a law of tangents[?]:

$\frac{a-b}{a+b} = \frac{\frac{\tan(A-B)}{2}} {\frac{\tan(A+B)}{2}}$

The trigonometric functions are also important outside of the study of triangles. They are periodic functions with characteristic wave patterns as graphs, useful for modelling recurring phenomena such as sound or light waves. Every signal can be written as a (typically infinite) sum of sine and cosine functions of different frequencies; this is the basic idea of Fourier analysis.

For a compilation of many relations between the trigonometric functions, see trigonometric identities.

An alternative use for trigonometric functions is to make pretty patterns.

{x, y} = Σ_n=1→∞   (1/F(n+1)){sin(θF(n)),cos(θF(n))}