 # 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 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