JH

Useful formulas

It's not always easy to find the formula you need, and impossible to remember them all, so here's a collection of some I have found useful.

sin A,   cos A
sin2A + cos2A = 1
sin2A = (1 - cos 2A)/2
sin A = 1 / cosec A & sin A = cos A tan A
sin (A+B) = sin A cos B + cos A sin B
sin (A-B) = sin A cos B - cos A sin B
sin 2A = 2 sin A cos A
sin A - sin B = 2 cos (A+B)/2 sin (A-B)/2
sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2
sin2A - sin2B = sin (A+B) sin (A-B)
Write tan θ/2 = t
... then  
sin θ = 2t / (1 + t2)
... and   cos θ = (1 - t2) / (1 + t2)
sinh x,   cosh x,   tanh x
  π radians = 180 degrees
  1 radian = 57.3 degrees
sinh x = (e x - e -x ) / 2
cosh x = (e x + e -x ) / 2
tanh x = sinh x / cosh x
cosh2x - sinh2x = 1
ex = sinh x + cosh x
log,   ln
log10 e = 0.43429
loge 10 = ln 10 = 2.30259
n log x = log xn
Roots of a quadratic
  If   y = a x2 + b x + c then ...
x = [ -b ± √( b2 - 4 a c)] / 2a   (2 roots)
Stray capacitance
Capacitance between 2 plates in air is ...
0.9 pF /sq.cm. /mm separation
Approximations
Provided that d <<1 ...
1 / (1 - d) ≈ 1 + d
1 / (1 + d) ≈ 1 - d
(1 ± d)n ≈ 1 ± nd
sin A,   cos A
sin, cos, tan
cos A = 1 / sec A & cos A = sin A / tan A
cos2A = (1 + cos 2A)/2
cos (A+B) = cos A cos B - sin A sin B
cos(A-B) = cos A cos B + sin A sin B
cos 2A = cos2A - sin2A
cos B - cos A = 2 sin (A+B)/2 sin (A-B)/2
1 - sin A = coversin A
cos2A - sin2B = cos (A+B) cos (A-B)
Complex numbers where j = √-1
(a + jb) = √ [a2 + b2]   tan-1(b/a)
e j θ = cos θ + j sin θ
e -j θ = cos θ - j sin θ
cos θ = (e j θ + e-j θ ) / 2
sin θ = (e j θ - e-j θ ) / 2j
e jnθ = cos nθ + j sin nθ
n (cos θ + j sin θ) = cos nθ + j sin nθ
Geometric progression
  If a series is a, ar, ar2, ar3, then ...
nth term = a r(n-1)
Sum of first n terms is S = a (rn - 1) / (r - 1)
Energy in a capacitor
If a capacitor C is charged to V, then
Energy stored (joules): J = C V2 / 2
Energy in an inductor
If an inductor L is carrying I amps, then
Energy stored (joules): J = L I2 / 2
Small angles
Provided that d (radians) is very small ...
sin d ≈ d   & sinh d ≈ d
cos d ≈ 1   & cosh d ≈ 1
tan d ≈ d   & tanh d ≈ d
tan A
tan A = sin A / cos A .. & .. cot A = 1 / tan A
1 + tan2A = sec2A .. & .. 1 + cot2A = cosec2A
tan2A = (1 - cos 2A) / (1 + cos 2A)
tan (A+B) = [tan A + tan B] / [1 - tan A tan B]
tan (A-B) = [tan A - tan B] / [1 + tan A tan B]
tan 2A = 2 tan A / (1 - tan2A)
tan (A/2) = sin A / (1 + cos A)
tan A + tan B = sin (A-B) / cos A cos B
tan A - tan B = sin (A+B) / cos A cos B
cot A + cot B = sin (A+B) / sin A sin B
cot A - cot B = sin (-A+B) / sin A sin B
Binomial theorem
(1 ± x)n = 1  ± nx  ± n (n-1) x2/ (1 . 2) ...
  ...  ± n (n-1) (n-2) x3/ (1 . 2 . 3) ... etc
e
e = 1 + 1 + 1/(1 . 2) + 1/(1 . 2 . 3) ...
  ...   + 1/(1 . 2 . 3 . 4) ... etc ...   = 2.71828
ex = 1 + x + x2/(1 . 2) ...
  ...   + x3/(1 . 2 . 3) + x4/(1 . 2 . 3 . 4) ... etc
Arithmetic progression
  If a series is a, (a+d), (a+2d), (a+3d), then ...
nth term = a + (n - 1) d
Sum of first n terms is
S = a n + (n - 1) n d / 2
Root-mean-square (RMS)
If a sinewave voltage has a peak value of ± E, then
Erms = E / √2
Mnemonic for π
How I Need A Drink, Alcoholic Of Course ...
( π = 3.1415926 ... )
Conversions
One pound (lb) = 454 grams
One mile = 1,760 yards = 5,280 ≈ 5,000 feet
One year = 8,760 ≈ 104 hours
A gallon of water weighs 10 lb. (in UK!)
One horse-power = 746 watts
One atmosphere = 14.7 psi = 1013 mb
One mile per hour = 1.467 ≈ 1.5 feet/sec