Thursday, February 17, 2011

Useful Mathematics

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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 &amp;mnplus; 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 &lt; x &amp;ltet; 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

Apple Opening Biggest Store To Date In Grand Central Station

Best Blogger Tips
It has been confirmed by a close source to the Cult of Mac website that plans for Apple to build a new store in Grand Central Station are indeed true and the store will be Apples biggest and grandest to date.
According to the source the new store already has the name Grand Central and that Apple will be making an internal announcement within the next month or so. The new Apple store is expected to open in the Autumn sometime in early September.
Apple Store
The unamed source also said that the new Grand Station Apple Store  will be a grand affair saying: “The company will certainly pull out all the stops on this one,”
Apple retail outlets have helped grow the company and with this year being the 10th anniversary of Apple retail, you can expected something spectacular from Apple in Grand Central Station.
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