Identities -- a compendium
of basic trig identities as follows:
co-function and phase shift identities
� exact values of
trig functions of common angles -- "Brain Dump", with another page for
identity: sin2x + cos2x
� half angle
� double angle
formulas, and another page for
Sin or Cos 3x, 4x, etc.,
and yet another page forContinued Fraction Tan.
etc. angle formulas
� double angle formulas expressed
in terms of
� 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
� generalized phase
identities -- sin(arccos(x)), etc.
identities -- based on eix=cos(x)+i
sin(x), such as cosh(ix) = cos(x),
� 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
-- 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
cos and sin product
identity -- cos(a−b) cos(t+u) − cos(a+b) cos(t−u) = 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))
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).
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.