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 Math Help > Trigonometry > Trig Equivalences

The Summary of Identities contains all of the basic trigonometric identities that you will need to use.  In addition,

### Contents of this section:

Identities -- a compendium of basic trig identities as follows:

even/odd, co-function and phase shift identities
exact values of trig functions of common angles -- "Brain Dump", with another page for more
Special Angles
Pythagorean identity: sin2x + cos2x = 1
half angle formulas
double angle formulas, and another page for
Sin or Cos 3x, 4x, etc., and yet another page forContinued Fraction Tan.
� double angle formulas expressed in terms of tan(x/2)
cos(x+y) etc. -- cos of sum, sin of sum, tan of sum formulas
cos(x) cos(y) -- sum of cos, etc. -- converting between a sum and a product of trig functions
generalized phase shift
"geometric progression" identities
"Triangle" identities -- sin(arccos(x)), etc.
Euler identities -- based on eix=cos(x)+i sin(x), such as
cosh(ix) = cos(x), etc.
arctan -- Tan-1(1/2) + Tan-1(1/5) + Tan-1(1/8) = π/4, and Gregory's Formula for Arctan, Machin's formula for π/4,
(and a whole lot more info at tan x+y = (tan x + tan y) / (1 - tan x tan y))

Solving trig equations -- an introduction to methods of solving equations containing trig functions

Sample problems -- a collection of contrived trig identities that are given as problems to high school students, along with their solutions and a discussion of the methods used to prove such identities.

Phase shift -- expressing the sum of two sine waves as a single phase-shifted and amplitude-adjusted sine wave.

cos and sin product identity -- cos(a−b) cos(t+u) − cos(a+b) cos(t−u) = sin(u+a) sin(b−t) − sin(u−a) sin(b+t)

Broken Calculator Puzzle -- Suppose you had a calculator that is broken so that the only keys that still work are the sin, cos, tan, sin-1, cos-1, and tan-1 buttons.  The display initially shows 0.  Given any positive rational number q, show that pressing some finite sequence of buttons will yield q.

sin (x/2) cos (x/2) = (1/2) sin x -- a beautiful geometric proof

Cos x + y = cos x cos y - sin x sin y -- a geometric proof

tan x+y = (tan x + tan y) / (1 - tan x tan y)

cos(2x) = (1-tan�(x))/(1+tan�(x))

Hyperbolic Functions

Hypergeometric Function

### Internet references

SOS Math: Table of Trigonometric Identities

Gottfried Wilhelm Leibniz (b. 1646, d. 1716) was a German philosopher, mathematician, and logician who is probably most well known for having invented the differential and integral calculus (independently of Sir Isaac Newton).

Mathworld article: Leibniz Series.

### Related Pages in this website

Special Angles

Sin or Cos 3x, 4x, etc. -- trig functions of any multiple of an angle.

d/dx (sin x) = cos x, in the calculus section of this website

Hyperbolic Functions -- sinh(x) and cosh(x), which, together with exp(x) and the circular functions sin(x) and cos(x) form a family of functions.

Table of Integrals -- derivations of various special integrals requires extensive use of the trig identities on this page.

The webmaster and author of this Math Help site is Graeme McRae.