Trigonometric functions

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, or equally generally, as solutions of certain differential equations. All four approaches will be presented below.

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.

• sine (sin)
• cosine (cos)
• tangent (tan = sin / cos)
• secant (sec = 1 / cos)
• cosecant (csc = 1 / sin)
• cotangent (cot = cos / sin)

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

Another function, which was important historically (and appeared in the earliest tables) but is now little-used, is the versed sine (versin = 1 − cos), also called the sagitta ("arrow").

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 h.
• 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/h.

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

cos(A) = adj/hyp = b/h.

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

tan(A) = opp/adj = a/b.

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:

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

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:

sec(A) = hyp/adj = h/b.

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:

cot(A) = adj/opp = b/a.

Mnemonics

There are a number of mnemonics for the above definitions, for example SOHCAHTOA (sounds like "soak a toe-a"). It reminds one that:

• SOH ... sin = opposite/hypotenuse
• CAH ... cos = adjacent/hypotenuse
• TOA ... tan = opposite/adjacent.

Many other such words and phrases have been contrived; for more, see: trigonometry mnemonics.

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 easily computed by hand, as in the following examples. (In fact, the sine, cosine and tangent of any integer multiple of three degrees can be found exactly by hand.)

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 sine, cosine and tangent of an angle of 45 degrees. Using the Pythagorean theorem, c = √(a² + b²) = √2. This is illustrated in the following figure:

Therefore,

$\sin \left(45^{\circ }\right)={1 \over {\sqrt {2}}}={{\sqrt {2}} \over 2}$

$\cos \left(45^{\circ }\right)={1 \over {\sqrt {2}}}={{\sqrt {2}} \over 2}$

$\tan \left(45^{\circ }\right)={{\sqrt {2}} \over {\sqrt {2}}}=1$

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

$\sin \left(30^{\circ }\right)={1 \over 2}$

$\cos \left(30^{\circ }\right)={{\sqrt {3}} \over 2}$

$\tan \left(30^{\circ }\right)={{\sqrt {3}} \over 3}$

and

$\sin \left(60^{\circ }\right)={{\sqrt {3}} \over 2}$

$\cos \left(60^{\circ }\right)={1 \over 2}$

$\tan \left(60^{\circ }\right)={\sqrt {3}}$

Unit-circle definitions

The six trigonometric 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. It also provides a single visual picture that encapsulates at once all the important triangles we have used so far.

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. Let a line making an angle of θ with the positive half of the x-axis intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The triangle in the graphic reveals the reason: the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = 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 lengths 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.

The period of the sine, cosine, secant, or cosecant is a full circle, i.e. 2π radians or 360 degrees; the period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.

trig functions from circle

All of the trigonometric functions can be
constructed geometrically in terms of

a unit circle centered at O.

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

$\tan \theta ={\frac {\sin \theta }{\cos \theta }}$

$\sec \theta ={\frac {1}{\cos \theta }}$

$\csc \theta ={\frac {1}{\sin \theta }}$

$\cot \theta ={\frac {\cos \theta }{\sin \theta }}$

Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (shown above right), and similar such geometric definitions were used historically. In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(&theta) is AC (half of the chord), a definition introduced in India (see below). cos(&theta) is the horizontal distance OC, and versin(&theta) = 1 − cos(&theta) is CD. tan(&theta) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(&theta) is another tangent segment, AF. sec(&theta) = OE and csc(&theta) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively. From these constructions, it is easy to see that the secant and tangent functions diverge for θ of π/2 (90 degrees) and that cosecant and cotangent diverge for θ of zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.)

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

$\sin \left(x\right)=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}$

$\cos \left(x\right)=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}$

Using the theory of Taylor series and the facts that the derivative of sine is cosine and the derivative of cosine is negative sine, one can show that these definitions are equivalent to the ones given above. These definitions are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.g. in Fourier series), since the theory of infinite series can be developed from the foundations of the real number system, independent of any geometric considerations. The differentiability and continuity of these functions is then established.

By allowing the arguments of the sine and cosine function to be complex, it can be shown that they are the imaginary and real parts of the complex exponential function, respectively. This relationship was first noted by Euler and the equation expressing the relationship is called Euler's formula. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. It is also shown that exponential processes are intimately linked to periodic behavior.

Definitions via differential equations

Both the sine and cosine functions satisfy the differential equation

$y\,''=-y$

i.e. each is the additive inverse of its own second derivative. Among the 2-dimensional space of solutions of this equation, the sine function is characterized as the one that satisfies the initial conditions y(0) = 0 and y′(0) = 1. The cosine is the one that satisfies the initial conditions y(0) = 1 and y′(0) = 0.

The tangent function satisfies the nonlinear differential equation

$y\,'=1+y^{2}$

and the initial conditions y(0) = 0 and y′(0) = 1.

Inverse functions

The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principle inverses are usually defined as:

${\begin{matrix}{\mbox{for}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},&y=\arcsin(x)&{\mbox{if and only if}}&x=\sin(y)\\{\mbox{for}}&0\leq y\leq \pi ,&y=\arccos(x)&{\mbox{if and only if}}&x=\cos(y)\\{\mbox{for}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},&y=\arctan(x)&{\mbox{if and only if}}&x=\tan(y)\\{\mbox{for}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},y\neq 0,&y=\operatorname {arccsc}(x)&{\mbox{if and only if}}&x=\csc(y)\\{\mbox{for}}&0\leq y\leq \pi ,y\neq {\frac {\pi }{2}},&y=\operatorname {arcsec}(x)&{\mbox{if and only if}}&x=\sec(y)\\{\mbox{for}}&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},y\neq 0,&y=\operatorname {arccot}(x)&{\mbox{if and only if}}&x=\cot(y)\\\end{matrix}}$

For inverse trigonometric functions, the notations sin^{− 1} and cos^{− 1} are often used for arcsin and arccos, etc. When this notation is used, the inverse functions are often confused with the multiplicative inverses of the functions. Our notation avoids such confusion.

These functions may also be defined by proving that they are antiderivatives of other functions. Then each function is uniquely determined by its value at a single point:

$\arcsin \left(x\right)=\int {\frac {1}{\sqrt {1-x^{2}}}}dx+C$

$\arccos \left(x\right)=\int {\frac {-1}{\sqrt {1-x^{2}}}}dx+C$

$\arctan \left(x\right)=\int {\frac {1}{1+x^{2}}}dx+C$

$\operatorname {arccsc} \left(x\right)=\int {\frac {-1}{x{\sqrt {x^{2}-1}}}}dx+C$

$\,\operatorname {arcsec} \left(x\right)=\int {\frac {1}{x{\sqrt {x^{2}-1}}}}dx+C$

$\operatorname {arccot} \left(x\right)=\int {\frac {-1}{x^{2}+1}}dx+C$

Note: arcsec can also mean 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 of the Pythagorean theorem:

$c^{2}=a^{2}+b^{2}-2ab\cos(C)$

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:

${\frac {a+b}{a-b}}={\frac {\tan[{\frac {1}{2}}(A+B)]}{\tan[{\frac {1}{2}}(A-B)]}}$

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, where trigonometric series are used to solve a variety of boundary-value problems in partial differential equations.

Functions based on sine and cosine can make pretty pictures.

The image on the right displays a two-dimensional graph based on such a summation of sines and cosines, illustrating the fact that arbitrarily complicated closed curves can be described by a Fourier series. Its equation is:

$(x(\theta ),\,y(\theta ))=\sum _{n=1}^{\infty }{\frac {1}{F(n+1)}}(\sin(\theta \,F(n)),\,\cos(\theta \,F(n)))$

where F(n) is the nth Fibonacci number.

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

An alternative use for trigonometric functions is to make pretty patterns such as Lissajous figures.

History

The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 B.C.), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century A.D.) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula sin(A/2)² = (1 − cos(A))/2, allowing him to create tables with any desired accuracy. Neither the tables of Hipparchus nor of Ptolemy have survived to the present day.

The next significant development of trigonometry was in India, in the works known as the Siddhantas (4th-5th century A.D.), which first defined the sine as the modern relationship between half an angle and half a chord. The Siddhantas also contained the earliest surviving tables of sine values (along with 1 − cos values), in 3.75-degree intervals from 0 to 90 degrees.

The Hindu works were later translated and expanded by the Arabs, who by the 10th century (in the work of Abu'l-Wefa) were using all six trigonometric functions, and had sine tables in 0.25-degree increments, to 8 decimal places of accuracy, as well as tables of tangent values.

Our modern word sine comes, via sinus ("bay" or "fold") in Latin, from a mistranslation of the Sanskrit jiva (or jya). jiva (originally called ardha-jiva, "half-chord", in the 6th century Aryabhatiya) was transliterated by the Arabs as jiba, but was confused for another word, jaib ("bay"), by European translators such as Robert of Chester and Gherardo of Cremona in Toledo in the 12th century, perhaps because jiba had been abbreviated jb in Arabic.

All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by the De triangulis omnimodus (A.D. 1464) of Regiomontanus (A.D. 1436-1476), as well as his later Tabulae directionum (which included the tangent function, unnamed).

The Opus palatinum de triangulis of Rheticus, a student of Copernicus, was the first to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in A. D. 1596.

The Introductio in analysin infinitorum (A.D. 1748) of Euler was primarily responsible for establishing the analytic treatment of trigonometric functions, defining them as infinite series and presenting "Euler's formula" e^ix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec..

Links and references

References

• Carl B. Boyer, A History of Mathematics, 2nd ed. (Wiley, New York, 1991).
• Eli Maor, Trigonometric Delights (Princeton Univ. Press, 1998).
• "Trigonometric functions", MacTutor History of Mathematics Archive.

See also

• Generating trigonometric tables
• Hyperbolic function
• Pythagoras
• Pythagorean theorem
• Trigonometric identity

External link

• Dave's Short Trig Course uses interactive java applets that are very helpful for learning about trig functions on the unit circle, as well as being useful more generally for students of trigonometry.