Trigonometric identity

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In mathematics, trigonometric identities are equalities involving trigonometric functions that are true for all values of the occurring variables. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common trick involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Notation: With trigonometric functions, we define functions sin², cos², etc., such that sin²(x) = (sin(x))². Often, sin⁻¹(x) is used to denote the inverse function. In this article, we prefer to write either arcsin(x) to indicate the inverse function, or csc(x) to indicate the multiplicative inverse.

Table of contents

1 Definitions
2 Periodicity, symmetry and shifts
3 From the Pythagorean theorem
4 Addition/subtraction theorems
5 Double-angle formulas
6 Multiple-angle formulas
7 Power-reduction formulas
8 Half-angle formulas
9 Products to sums
10 Sums to products
11 Inverse trigonometric functions
12 The Gudermannian function
13 Identities with no variables
14 Calculus
15 Proofs using a differential equation
16 Geometric proofs

16.1 sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
16.2 cos(x + y) cos(x) cos(y) − sin(x) sin(y)

17 Abstract point of view
18 See also

Definitions

Periodicity, symmetry and shifts

These are most easily shown from the unit circle:

For some purposes it is important to know that any linear combination of sine waves of the same period but different phase shifts is also a sine wave with the same period, but a different phase shift. In other words, we have

where

From the Pythagorean theorem

Addition/subtraction theorems

The quickest way to prove these is Euler's formula. The tangent formula follows from the other two. A geometric proof of the sin(x + y) identity is given at the end of this article.

where

Double-angle formulas

These can be shown by substituting x = y in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivre's formula with n = 2.

Multiple-angle formulas

If T_n is the nth Chebyshev polynomial then

De Moivre's formula:

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 nth-degree Fourier approximation. The same holds for any measure or generalized function.

Power-reduction formulas

Solve the third and fourth double angle formula for cos²(x) and sin²(x).

Half-angle formulas

Substitute x/2 for x in the power reduction formulas, then solve for cos(x/2) and sin(x/2).

Multiply tan(x/2) by 2cos(x/2) / ( 2cos(x/2)) and substitute sin(x/2) / cos(x/2) for tan(x/2). The numerator is then sin(x) via the double-angle formula, and the denominator is 2cos²(x/2) − 1 + 1, which is cos(x) + 1 by the double-angle formulae. The second formula comes from the first formula multiplied by sin(x) / sin(x) and simplified using the Pythagorean trigonometric identity.

If we set

then

and

This substitution of t for tan(x/2), with the consequent replacement of sin(x) by 2t/(1 + t²) and cos(x) by (1 − t²)/(1 + t²) is useful in calculus for converting rational functions in in sin(x) and cos(x) to functions of t in order to find their antiderivatives. (See "abstract point of view" below.)

Products to sums

These can be proven by expanding their right-hand-sides using the addition theorems.

Sums to products

Replace x by (x + y) / 2 and y by (x – y) / 2 in the Product-to-Sum formulas.

Inverse trigonometric functions

The Gudermannian function

The Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers -- see that article for details.

Identities with no variables

Richard Feynman is reputed to have learned as a boy, and always remembered, the following curious identity:

However, this identity is a special case of one that does contain a variable:

The following are perhaps not as readily generalized to identities with variables in them:

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 the integers less than 21/2 that 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 MÃÂ¶bius 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.

The following identity with no variables can be used to compute &pi efficiently:

or by using Euler's formula:

Calculus

In calculus it is essential that angles that are arguments to trigonometric functions be measured in radians; if they are measured in degrees or any other units, then the relations stated below fail. If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by first verifying two limits.

(verified using the unit circle & squeeze theorem). It may be tempting to propose to use L'HÃÂ´pital's rule to establish this limit. However, if one uses this limit in order to prove that the derivative of the sine is the cosine, and then uses the fact that the derivative of the sine is the cosine in applying L'HÃÂ´pital's rule, one is reasoning circularly — a logical fallacy.

(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′ = cos and cos′ = −sin. If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.

The rest of the trigonometric functions can be differentiated using the above identities and the rules of differentiation. We have:

The integral identities can be found in Wikipedia's table of integrals.

Proofs using a differential equation

Consider this differential equation:

Using Euler's formula and the method for solving linear differential equations combined with the uniqueness theorem and the existence theorem we can define sine and cosine as the following1

is the unique solution of

subject to the initial conditions of and

is the unique solution of

subject to the initial conditions of and

Now, let's prove that

First let

Now we find the first and second derivatives of T(x)

but since is a solution of we can say so

Therefore

Therefore we can say

Once again according to our technique of solving linear differential equations and Euler's formula the solution to

must be a linear combination of and , therefore

Now we solve for B by plugging in 0 for x

but according to our initial values , therefore

To solve for A we take the derviative of T(x) and plug in 0 for x

Using our initial values and since

Plugging A and B back into our original equation for T(x) we get

But since T(x) was defined as we get

Q.E.D.

Using these rigorous definitions of sine and cosine, you can prove all the other properties of sine and cosine using the same technique.

Geometric proofs

sin(x + y) sin(x) cos(y) + cos(x) sin(y)

In the figure the angle x is part of right angled triangle ABC, and the angle y part of right angled triangle ACD. Then construct DG perpendicular to AB and construct CE parallel to AB.

Angle x = Angle BAC = Angle ACE = Angle CDE.

EG = BC.

;

cos(x + y) cos(x) cos(y) − sin(x) sin(y)

Using the above figure:

;

Abstract point of view

Since the circle is an algebraic curve of genus 0, one expects the 'circular functions' to be reducible to rational functions. This is known classically, by systematically using the tan-half-angle formulae to write the sine and cosine functions in terms of a new variable t.