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.

Table of contents

1 From the 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 Abstract point of view
16 A geometric proof of the identity of sin(x + y) and sin(x) cos(y) + cos(x) sin(y)

From the 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 in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivre's formula with .

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 square-integrable function of period 2π with the Dirichlet kernel coincides with the function's nth-degree Fourier approximation.

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 trig identity.

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 MÃÂÃÂ¶bius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities precedeing 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:

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 that

(verified using the unit circle and squeeze theorem)

and

(verified using the identity tan(x/2) = (1 − cos(x))/sin(x))

and then using the limit definition of the derivative and the addition theorems; if they are defined by their Taylor series, then the derivatives can be found by differentiating the power series term by term.

The rest of the trig functions can be differentiated using the above identities and the rules of differentiation, for instance

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

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.

A geometric proof of the identity of sin(x + y) and 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.

= (BC / AD) + (DE / AD)

= ((AC / AC) * (BC / AD)) + ((CD / CD) * (DE / AD))

= ((BC / AC) * (AC / AD)) + ((DE / CD) * (CD / AD))

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