Trigonometric function
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)
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).
Right triangle definitions
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.
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.
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.
- tan(A) = opp/adj = a/b.
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.
- sec(A) = hyp/adj = h/b.
- 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.
Computing
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 = √(a^{2} + b^{2}) = √2. This is illustrated in the following figure:
Therefore,
See Also
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:
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:
The smallest positive period of a periodic function is called the primitive period of the function. The primitive period of the sine, cosine, secant, or cosecant is a full circle, i.e. 2π radians or 360 degrees; the primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.
Above, only sine and cosine were defined directly by the unit circle, but the other four trig functions can be defined by:
All of the trigonometric functions can be constructed geometrically in terms of a unit circle centered at O
Here is a plot of sine and cosine:
Series definitions
The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin
Please note: Here, and generally in calculus, it is of utmost importance that all angles are measured in radians.
Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is negative sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x:
Relationship to exponential function
By allowing the arguments of the sine and cosine functions to be complex, it can be shown that they are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary:
Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:
Definitions via differential equations
Both the sine and cosine functions satisfy the differential equation
The tangent function is the unique solution of the nonlinear differential equation
Inverse functions
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:
Identities
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:
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:
There is also a law of tangents:
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 curvess can be described by a Fourier series. Its equation is:
For a compilation of many relations between the trigonometric functions, see trigonometric identities.
History
The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), 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) 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), 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 (1464) of Regiomontanus (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 1596.
The Introductio in analysin infinitorum (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..
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.
- Tristan Needham, Visual Complex Analysis, (Oxford University Press, 2000), ISBN 0198534469 Book website
See also
- Generating trigonometric tables
- Hyperbolic function
- Pythagoras
- Pythagorean theorem
- Trigonometric identity
- What is trigonometry used for
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.