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Trigonometry 

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In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of nontrigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
This article uses Greek letters such as alpha (α), beta (β), gamma (γ), and theta (θ) to represent angles. Several different units of angle measure are widely used, including degrees, radians, and grads:
The following table shows the conversions for some common angles:
Degrees  30°  60°  120°  150°  210°  240°  300°  330° 

Radians  
Grads  33⅓ grad  66⅔ grad  133⅓ grad  166⅔ grad  233⅓ grad  266⅔ grad  333⅓ grad  366⅔ grad 
Degrees  45°  90°  135°  180°  225°  270°  315°  360° 
Radians  
Grads  50 grad  100 grad  150 grad  200 grad  250 grad  300 grad  350 grad  400 grad 
Unless otherwise specified, all angles in this article are assumed to be in radians, but angles ending in a degree symbol (°) are in degrees. Per Niven's theorem multiples of 30° are the only angles that are a rational multiple of one degree and also have a rational sin/cos, which may account for their popularity in examples.^{[1]}
The primary trigonometric functions are the sine and cosine of an angle. These are sometimes abbreviated sin(θ) and cos(θ), respectively, where θ is the angle, but the parentheses around the angle are often omitted, e.g., sin θ and cos θ.
The sine of an angle is defined in the context of a right triangle, as the ratio of the length of the side that is opposite to the angle divided by the length of the longest side of the triangle (the hypotenuse ).
The cosine of an angle is also defined in the context of a right triangle, as the ratio of the length of the side the angle is in divided by the length of the longest side of the triangle (the hypotenuse ).
The tangent (tan) of an angle is the ratio of the sine to the cosine:
Finally, the reciprocal functions secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent:
These definitions are sometimes referred to as ratio identities.
The inverse trigonometric functions are partial inverse functions for the trigonometric functions. For example, the inverse function for the sine, known as the inverse sine (sin^{−1}) or arcsine (arcsin or asin), satisfies
and
This article uses the notation below for inverse trigonometric functions:
Function  sin  cos  tan  sec  csc  cot 

Inverse  arcsin  arccos  arctan  arcsec  arccsc  arccot 
In trigonometry, the basic relationship between the sine and the cosine is known as the Pythagorean identity:
where cos^{2} θ means (cos(θ))^{2} and sin^{2} θ means (sin(θ))^{2}.
This can be viewed as a version of the Pythagorean theorem, and follows from the equation x^{2} + y^{2} = 1 for the unit circle. This equation can be solved for either the sine or the cosine:
where the sign depends on the quadrant of θ.
Dividing the Pythagorean identity by either cos^{2} θ or sin^{2} θ yields two other identities:
Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):
in terms of  

The versine, coversine, haversine, and exsecant were used in navigation. For example the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.
Name(s)  Abbreviation(s)  Value^{[3]} 

versed sine, versine  
versed cosine, vercosine  
coversed sine, coversine  
coversed cosine, covercosine  
half versed sine, haversine  
half versed cosine, havercosine  
half coversed sine, hacoversine cohaversine  
half coversed cosine, hacovercosine cohavercosine  
exterior secant, exsecant  
exterior cosecant, excosecant  
chord 
By examining the unit circle, the following properties of the trigonometric functions can be established.
When the trigonometric functions are reflected from certain angles, the result is often one of the other trigonometric functions. This leads to the following identities:
Reflected in ^{[4]}  Reflected in (cofunction identities)^{[5]}  Reflected in 

Note that the sign in front of the trig function does not necessarily indicate the sign of the value. For example, does not always mean that is positive. In particular, if , then .
By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express particular results more simply. Some examples of this are shown by shifting functions round by π/2, π and 2π radians. Because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift.
Shift by π/2  Shift by π Period for tan and cot^{[6]}  Shift by 2π Period for sin, cos, csc and sec^{[7]} 

These are also known as the addition and subtraction theorems or formulae. They were originally established by the 10th century Persian mathematician Abū alWafā' Būzjānī. One method of proving these identities is to apply Euler's formula. The use of the symbols and is described in the article plusminus sign.
For the angle addition diagram for the sine and cosine, the line in bold with the 1 on it is of length 1. It is the hypotenuse of a right angle triangle with angle β which gives the sin β and cos β. The cos β line is the hypotenuse of a right angle triangle with angle α so it has sides sin α and cos α both multiplied by cos β. This is the same for the sin β line. The original line is also the hypotenuse of a right angle triangle with angle α+β, the opposite side is the sin(α+β) line up from the origin and the adjacent side is the cos(α+β) segment going horizontally from the top left.
Overall the diagram can be used to show the sine and cosine of sum identities
because the opposite sides of the rectangle are equal.
Sine  ^{[8]}^{[9]} 

Cosine  ^{[9]}^{[10]} 
Tangent  ^{[9]}^{[11]} 
Arcsine  ^{[12]} 
Arccosine  ^{[13]} 
Arctangent  ^{[14]} 
The sum and difference formulae for sine and cosine can be written in matrix form as:
This shows that these matrices form a representation of the rotation group in the plane (technically, the special orthogonal group SO(2)), since the composition law is fulfilled: subsequent multiplications of a vector with these two matrices yields the same result as the rotation by the sum of the angles.
In these two identities an asymmetry appears that is not seen in the case of sums of finitely many terms: in each product, there are only finitely many sine factors and cofinitely many cosine factors.
If only finitely many of the terms θ_{i} are nonzero, then only finitely many of the terms on the right side will be nonzero because sine factors will vanish, and in each term, all but finitely many of the cosine factors will be unity.
Let e_{k} (for k = 0, 1, 2, 3, ...) be the kthdegree elementary symmetric polynomial in the variables
for i = 0, 1, 2, 3, ..., i.e.,
Then
The number of terms on the right side depends on the number of terms on the left side.
For example:
and so on. The case of only finitely many terms can be proved by mathematical induction.^{[15]}
where e_{k} is the kthdegree elementary symmetric polynomial in the n variables x_{i} = tan θ_{i}, i = 1, ..., n, and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left. The case of only finitely many terms can be proved by mathematical induction on the number of such terms. The convergence of the series in the denominators can be shown by writing the secant identity in the form
and then observing that the left side converges if the right side converges, and similarly for the cosecant identity.
For example,
T_{n} is the nth Chebyshev polynomial  ^{[16]} 

S_{n} is the nth spread polynomial  
de Moivre's formula, is the imaginary unit  ^{[17]} 
These can be shown by using either the sum and difference identities or the multipleangle formulae.
Doubleangle formulae^{[18]}^{[19]}  

Tripleangle formulae^{[16]}^{[20]}  
Halfangle formulae^{[21]}^{[22]}  
The fact that the tripleangle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that this is in general impossible using the given tools, by field theory.
A formula for computing the trigonometric identities for the thirdangle exists, but it requires finding the zeroes of the cubic equation , where x is the value of the sine function at some angle and d is the known value of the sine function at the triple angle. However, the discriminant of this equation is negative, so this equation has three real roots (of which only one is the solution within the correct thirdcircle) but none of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots, (which may be expressed in terms of realonly functions only if using hyperbolic functions).
For specific multiples, these follow from the angle addition formulas, while the general formula was given by 16th century French mathematician Vieta.
In each of these two equations, the first parenthesized term is a binomial coefficient, and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed. tan nθ can be written in terms of tan θ using the recurrence relation:
cot nθ can be written in terms of cot θ using the recurrence relation:
The Chebyshev method is a recursive algorithm for finding the n^{th} multiple angle formula knowing the (n − 1)^{th} and (n − 2)^{th} formulae.^{[23]}
The cosine for nx can be computed from the cosine of (n − 1)x and (n − 2)x as follows:
Similarly sin(nx) can be computed from the sines of (n − 1)x and (n − 2)x
For the tangent, we have:
where H/K = tan(n − 1)x.
Setting either α or β to 0 gives the usual tangent halfangle formulæ.
(Refer to sinc function.)
Obtained by solving the second and third versions of the cosine doubleangle formula.
Sine  Cosine  Other 

and in general terms of powers of sin θ or cos θ the following is true, and can be deduced using De Moivre's formula, Euler's formula and binomial theorem^{[citation needed]}.
Cosine  Sine  

The producttosum identities or prosthaphaeresis formulas can be proven by expanding their righthand sides using the angle addition theorems. See amplitude modulation for an application of the producttosum formulæ, and beat (acoustics) and phase detector for applications of the sumtoproduct formulæ.


Charles Hermite demonstrated the following identity.^{[26]} Suppose a_{1}, ..., a_{n} are complex numbers, no two of which differ by an integer multiple of π. Let
(in particular, A_{1,1}, being an empty product, is 1). Then
The simplest nontrivial example is the case n = 2:
(The first three equalities are trivial; the fourth is the substance of this identity.) Essentially this is Ptolemy's theorem adapted to the language of modern trigonometry.
For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the a and b unknowns of the inphase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of c and φ. In the case of a nonzero linear combination of a sine and cosine wave^{[dead link]}^{[27]} (which is just a sine wave with a phase shift of π/2), we have
where
and (using the atan2 function)
More generally, for an arbitrary phase shift, we have
where
and
The general case reads^{[citation needed]}
where
and
See also Phasor addition.
These identities, named after Joseph Louis Lagrange, are:^{[28]}^{[29]}
A related function is the following function of x, called the Dirichlet kernel.
Sum of sines and cosines with arguments in arithmetic progression:^{[30]} if , then
For any a and b:
where atan2(y, x) is the generalization of arctan(y/x) that covers the entire circular range.
The above identity is sometimes convenient to know when thinking about the Gudermannian function, which relates the circular and hyperbolic trigonometric functions without resorting to complex numbers.
If x, y, and z are the three angles of any triangle, i.e. if x + y + z = π, then
If ƒ(x) is given by the linear fractional transformation
and similarly
then
More tersely stated, if for all α we let ƒ_{α} be what we called ƒ above, then
If x is the slope of a line, then ƒ(x) is the slope of its rotation through an angle of −α.
and hence the corollary:
where .
For applications to special functions, the following infinite product formulae for trigonometric functions are useful:^{[34]}^{[35]}
The curious identity
is a special case of an identity that contains one variable:
Similarly:
The same cosine identity in radians is
Similarly:
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
Many of those curious identities stem from more general facts like the following:^{[36]}
and
Combining these gives us
If n is an odd number (n = 2m + 1) we can make use of the symmetries to get
The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:
An efficient way to compute π is based on the following identity without variables, due to Machin:
or, alternatively, by using an identity of Leonhard Euler:
For certain simple angles, the sines and cosines take the form for 0 ≤ n ≤ 4, which makes them easy to remember.
With the golden ratio φ:
Also see exact trigonometric constants.
Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:
Ptolemy used this proposition to compute some angles in his table of chords.
This identity involves a trigonometric function of a trigonometric function:^{[37]}
where J_{0} and J_{2k} are Bessel functions.
In calculus the relations stated below require angles to be measured in radians; the relations would become more complicated if angles were measured in another unit such as degrees. If the trigonometric functions are defined in terms of geometry, along with the definitions of arc length and area, their derivatives can be found by verifying two limits. The first is:
verified using the unit circle and squeeze theorem. The second limit is:
verified using the identity tan(x/2) = (1 − cos x)/sin x. Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that (sin x)′ = cos x and (cos x)′ = −sin x. If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series termbyterm.
The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation:^{[38]}^{[39]}^{[40]}
The integral identities can be found in "list of integrals of trigonometric functions". Some generic forms are listed below.
The fact that the differentiation of trigonometric functions (sine and cosine) results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and Fourier transforms.
Let i = √−1 be the imaginary unit and let denote composition of differential operators. Then for every odd positive integer n,
(When k = 0, then the number of differential operators being composed is 0, so the corresponding term in the sum above is just (sin x)^{n}.) This identity was discovered as a byproduct of research in medical imaging.^{[41]}
Function  Inverse function^{[42]} 

The Dirichlet kernel D_{n}(x) is the function occurring on both sides of the next identity:
The convolution of any integrable function of period 2π with the Dirichlet kernel coincides with the function's nthdegree Fourier approximation. The same holds for any measure or generalized function.
If we set
then^{[43]}
where e^{ix} = cos(x) + i sin(x), sometimes abbreviated to cis(x).
When this substitution of t for tan(x/2) is used in calculus, it follows that sin(x) is replaced by 2t/(1 + t^{2}), cos(x) is replaced by (1 − t^{2})/(1 + t^{2}) and the differential dx is replaced by (2 dt)/(1 + t^{2}). Thereby one converts rational functions of sin(x) and cos(x) to rational functions of t in order to find their antiderivatives.