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Useful Mathematics

Algebra

The Binomial Theorem

(1 + x)ⁿ = 1 + nx + [n(n-1) x²]/(2!) + [n(n-1)(n-2) x³]/(3!) + ...

If x << 1, then

(1 + x)ⁿ ≅ 1 + n x

(1 + x)^-n ≅ 1 - n x

These approximations are useful when x² is negliable.

Quadratic Equations

ax² + bx + c = 0 has the solution,

x ={[-b ± (b² - 4ac)]^1/2} / (2a)

Trigonometry

π rad = 180 °

1 rad = 57.3 °

The quadrants in which trigonometrical functions are positive. Is shown below:

Figure 1. Signs of trigonometric functions.

A good way to remember this is the phrase clockwise ACTS. Clockwise gives the direction from the first quadrant is clockwise and each letter from the word ACTS stands for a trigonometric function: All, Cos, Tan and Sin. The direction of the angle increases in an anti-clockwise sense.

If A and B are angles then

tan A = sin A/cos A

sin² A + cos² A = 1

sec²A = 1 + tan² A

cosec² A = 1 + cot² A

sin (A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B -/+; sin A sin B

tan (A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

If t= tan (1/2) A, sin A = (2t) / (1 + t²), cos A = (1 - t²) / (1 + t²)

2 sin A cos B = sin (A + B) + sin (A - B)

2 cos A cos B = cos (A + B) + cos (A - B)

2 sin A sin B = cos (A - B) - cos (A + B)

sin A + sin B = 2 sin [(A + B)/2] cos [(A - B)/2]

sin A - sin B = 2 cos [(A + B)/2] sin [(A - B)/2]

cos A + cos B = 2 cos [(A + B)/2] cos [(A - B)/2]

cos A - cos B = 2 sin [(A + B)/2] sin [(A - B)/2]

Power Series

e^x = exp x = 1 + x + x²/(2!) + ... + x^r/(r!) + ... for all x

ln (1 + x) = x - x²/ 2 + x³/3 - ... + (-1)^r+1x^r/r + ... (-1 < x <et; 1)

cos x = (e^ix + e^-ix)/2 = 1 - x²/(2!) + x⁴/(4!) - ... + (-1)^rx^2r/(2r)! + ... for all x

sin x = (e^ix - e^-ix)/(2i) = x - x³/(3!) + x⁵/(5!) - ... + (-1)^rx^2r+1/(2r + 1)! + ... for all x

cosh x = (e^x + e^-x)/2 = 1 + x²/(2!) + x⁴/(4!) + ... + x^2r/(2r)! + ... for all x

sinh x = (e^x - e^-x)/2 = x + x³/(3!) + x⁵/(5!) + ... + x^2r+1/(2r + 1)! + ... for all x