Trigonometric Identities

Trigonometric identities — mathematical expressions for trigonometric functions that hold for all values of the argument (from the general domain).

Basic trigonometric formulas

№	Formula	Permissible values of the argument
1.1
1.2
1.3
1.4

• Formula (1.1) is a corollary of the Pythagorean theorem.
• Formulas (1.2) and (1.3) are obtained from formula (1.1) by dividing by and respectively.
• Formula (1.4) follows from the definitions of tangent and cotangent.

Remark

There are other trigonometric functions as well.

Angle addition and subtraction formulas

№	Angle addition and subtraction formulas
2.1
2.2
2.3
2.4

Formula (2.3) is obtained by dividing (2.1) by (2.2). And formula (2.4) — by dividing (2.2) by (2.1).

Double-angle and half-angle formulas

The double-angle formulas are derived from formulas (2.1)—(2.4) , if we take the angle β to equal the angle α:

№	Double-angle formulas
3.1
3.2
3.3
3.4

Notes[show]

From the double-angle formula for cosine (3.2) the half-angle formulas are derived:

№	Half-angle formulas
3.5
3.6
3.7

Triple-angle formulas

The triple-angle formulas are derived from formulas (2.1)—(2.4) , if we take the angle β to equal the angle 2α:

№	Triple-angle formulas
4.1
4.2
4.3
4.4

Power-reduction formulas

The power-reduction formulas are derived from formulas (3.2):

№	Sine		№	Cosine
5.1		5.5
5.2		5.6
5.3		5.7
5.4		5.8

№	Product
5.9
5.10
5.11
5.12

Formulas for converting a product of functions

№	Formulas for converting products of functions
6.1
6.2
6.3

Formulas for converting a sum of functions

№	Formulas for converting a sum of functions
7.1
7.2
7.3
7.4
7.5

Derivation of formulas for converting a sum of functions[show]

Conversion of a sum of sines of 3 different angles into a product when : :

(7.6)

Solving simple trigonometric equations

If — there are no real solutions.

If — the solution is a number of the form

If — there are no real solutions.

If — the solution is a number of the form

The solution is a number of the form

The solution is a number of the form

Universal trigonometric substitution

The identities make sense only when both sides exist (that is, when ).

Auxiliary argument (formulas for the addition of harmonic oscillations)

The sum of two harmonic oscillations of the same frequency will again be a harmonic oscillation. In particular,

where , and are not simultaneously equal to zero, — is an angle called the auxiliary argument, which can be found from the system of equations:

Note. From the system above it follows that , however it cannot always be assumed that . The signs of and must be taken into account to determine which quadrant the angle belongs to.

Useful identities

In the formulas below, the numbers and are integers.

The following formula is given in two variants for the angle given in degrees and radians:

Representation of trigonometric functions in complex form

Euler's formula states that for any real number the following equality holds:

where — is the base of the natural logarithm,

— is the imaginary unit.

Using Euler's formula, the functions and can be defined as follows:

It follows that

See also

• Hyperbolic functions
• Sine integral
• Cosine integral
• Complex numbers
• Chebyshev polynomials
• Inverse trigonometric functions
• Rarely used trigonometric functions
• Solving triangles
• Versine
• Spherical trigonometry
• Triangle § Trigonometric identities involving only angles
• Trigonometric functions
• Trigonometric functions of a matrix
• Trigonometric Fourier series
• Gudermannian function
• Four-figure mathematical tables (Bradis tables)
• Elliptic functions