List of trigonometric identities

2.1 Mutual representation
2.2 sign of the angular functions
2.3 Important function values
2.4 symmetries
2.5 phase shifts
2.6 Traceability to acute angles
2.7 Presentation by the tangent of the half angle
Addition theorems 2.8
Addition theorems for 2.9 Inverse Trigonometric Functions
2:10 Double trigonometric functions
2:11 trigonometric functions for other multiples
2:12 half-angle formulas
2:13 sums of two trigonometric functions ( identities )
2:14 products of trigonometric functions
2:15 powers of trigonometric functions 2.15.1 sine
2.15.2 cosine
2.15.3 tangent

Triangulation

The following list contains the most known formulas of trigonometry in the plane. Most of these relationships using trigonometric functions.

The following designations are used: The triangle ABC was the pages A = BC, CA, and b = c = a, the angles α, β and γ in the corners A, B and C. Further, the radius r be the radius, ρ the Inkreisradius and ρa, ρb and ρc the Ankreisradien (and the radii of the excircles that the vertices A, B and C are opposite ) of the triangle ABC. The variable s is half the perimeter of the triangle ABC. Finally, the area of triangle ABC is referred to as F. All other terms are explained respectively in the corresponding sections in which they occur.

Angle sum

Law of Sines

Formula 1:

Formula 2: if

Law of cosines

Formula 1:

Formula 2: if

Projection set

The Mollweide 's formulas

Tangent set

Formula 1:

Analogous formulas are valid for (c a) / ( c - a ) and (a b ) / (a - b).

Formula 2: if α = 90 °

Formulas with half the circumference

S is always below the half of the perimeter of the triangle ABC, that is.

Area and circumradius

The area of the triangle ABC is here denoted by F (not as common today, with A in order to avoid any confusion with the triangular corner A):

The circumradius of triangle ABC we denote by r.

( It should be noted that as used herein, names r, ρ, ρa, ρb, ρc for the circumradius, the Inkreisradius and the three Ankreisradien of the mainly popular in the English-speaking notation differ, in which the same variables R, r, ra, rb, rc are called. )

Heron's formula:

Extended Law of Sines:

In and Ankreisradien

In this section, formulas are listed in which the Inkreisradius ρ and the Ankreisradien ρa, ρb and ρc occur of triangle ABC.

Important inequality; Equality occurs only if triangle ABC is equilateral.

The excircles are equal: Each formula is valid for ρa in analog form for ρb and ρc.

The lengths of the starting from A, B and C, respectively heights of triangle ABC are with ha, hb and hc respectively.

Has a right-angle triangle ABC with C ( is therefore γ = 90 °), the following applies

Seitenhalbierende

The lengths of the starting from A, B or C medians of the triangle ABC are called sa, sb and sc.

Bisector

We denote by wα, wβ and wγ the lengths of the starting from A, B or C bisector in triangle ABC.

General Trigonometry in the plane

Mutual representation

The trigonometric functions can be converted into each other or be mutually exclusive. There are the following relationships:

( See also phase shifts. )

By means of these equations can be the three functions occurring through one of the other two represent:

Sign of the angular functions

The sign of cot, sec and csc are consistent with those of their reciprocal functions tan, cos and sin.

Important function values

Weblink: more values

Symmetries

The trigonometric functions have simple symmetries:

Phase shifts

Return on acute angles

Representation by the tangent of the half angle

With the designation, the following relations are valid for any

Addition theorems

For any follow the double-angle functions for the phase shifts.

Addition theorems for Inverse Trigonometric Functions

For the Inverse Trigonometric Functions following addition theorems apply

Double trigonometric functions

Trigonometric functions for other multiples

The formula for standing over the Chebyshev polynomials in relationship.

Half-angle formulas

To calculate the function value of the half argument serve the half-angle formulas. The sign changes every 360 ° ( and ) or 180 ° for all and.

Also applies to a limited range of x:

Sum of two trigonometric functions ( identities )

From the addition theorems to identities can be derived with the help of the sum of two trigonometric functions can be represented as a product:

It still give special cases:

Products of the trigonometric functions

Products of the trigonometric functions can be calculated with the following formulas:

From the double angle feature also follows:

Powers of trigonometric functions

Sine

Cosine

Tangent

Conversion to other trigonometric functions

Other formulas for the case α β γ = 180 °

The following formulas hold for arbitrary planar triangles and follow longer term transformations from α β γ = 180 °, while the characters appearing in the formulas functions are well defined (the latter applies only to the formulas in which tangent and cotangent occur ).

Sinusoid and linear combination with the same phase

In which

Is generally

In which

And

Series expansion

As elsewhere in calculus, all angles are in radians.

It can be shown that represents the derivative of the cosine and sine is the derivative of the cosine of a negative sine wave. If you have these recordings, one can develop the Taylor series (easiest with the development of the point x = 0) and show that the following identities for all x are from the real numbers. With this series, the trigonometric functions for complex arguments are defined (or denotes the Bernoulli numbers ):