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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.

ax2 + bx + c = 0 has the solution,
x ={[-b ± (b2 - 4ac)]1/2} / (2a)

### Trigonometry

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

ex = exp x = 1 + x + x2/(2!) + ... + xr/(r!) + ... for all x
ln (1 + x) = x - x2/ 2 + x3/3 - ... + (-1)r+1xr/r + ... (-1 < x ..
cos x = (eix + e-ix)/2 = 1 - x2/(2!) + x4/(4!) - ... + (-1)rx2r/(2r)! + ... for all x
sin x = (eix - e-ix)/(2i) = x - x3/(3!) + x5/(5!) - ... + (-1)rx2r+1/(2r + 1)! + ... for all x
cosh x = (ex + e-x)/2 = 1 + x2/(2!) + x4/(4!) + ... + x2r/(2r)! + ... for all x
sinh x = (ex - e-x)/2 = x + x3/(3!) + x5/(5!) + ... + x2r+1/(2r + 1)! + ... for all x

..

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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.

ax2 + bx + c = 0 has the solution,

x ={[-b ± (b2 - 4ac)]1/2} / (2a)

### Trigonometry

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

ex = exp x = 1 + x + x2/(2!) + ... + xr/(r!) + ... for all x

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

cos x = (eix + e-ix)/2 = 1 - x2/(2!) + x4/(4!) - ... + (-1)rx2r/(2r)! + ... for all x

sin x = (eix - e-ix)/(2i) = x - x3/(3!) + x5/(5!) - ... + (-1)rx2r+1/(2r + 1)! + ... for all x

cosh x = (ex + e-x)/2 = 1 + x2/(2!) + x4/(4!) + ... + x2r/(2r)! + ... for all x

sinh x = (ex - e-x)/2 = x + x3/(3!) + x5/(5!) + ... + x2r+1/(2r + 1)! + ... for all x

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Дэйв Скотт бросает молоток и перо, чтобы проверить теорию падения объектов под действием силы тяжести.

Dave Scott drops a hammer and a feather to test the theory of falling objects in a gravity well

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