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: sin^{2}x + cos^{2}x
= 1
� half angle
formulas
� double angle
formulas, and another page for
Sin or Cos 3x, 4x, etc.,
and yet another page forContinued Fraction Tan.
�
triple, quadruple,
etc. angle formulas
� 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 e^{ix}=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
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.
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.