Derivation of Formula | MATHalino reviewer about Derivation of Formula

The formula for the radius of the circle circumscribed about a triangle (circumcircle) is given by

$R = \dfrac{abc}{4A_t}$

where A_t is the area of the inscribed triangle.

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The radius of incircle is given by the formula

$r = \dfrac{A_t}{s}$

where A_t = area of the triangle and s = semi-perimeter.

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The following are the formulas for cosine law for any triangles with sides a, b, c and angles A, B, C, respectively.

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

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For any triangles with vertex angles and corresponding opposite sides are A, B, C and a, b, c, respectively, the sine law is given by the formula...

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

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Triangle used in sum and difference of two angles The sum and difference of two angles can be derived from the figure shown below.

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The depreciation charge and the total depreciation at any time m using the sum-of-the-years-digit method is given by the following formulas:

Depreciation Charge:

$d_m = (FC - SV) \dfrac{n - m + 1}{SYD}$

Total depreciation at any time m

$D_m = (FC - SV) \dfrac{m(2n - m + 1)}{2 \times SYD}$

Where:
FC = first cost
SV = salvage value
n = economic life (in years)
m = any time before n (in years)
SYD = sum of years digit = 1 + 2 + ... + n = n(1 + n)/2

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Frustum of a pyramid and frustum of a cone

The formula for frustum of a pyramid or frustum of a cone is given by

$V = \dfrac{h}{3} \left[ \, A_1 + A_2 + \sqrt{A_1A_2} \, \right]$

Where:
h = perpendicular distance between A₁ and A₂ (h is called the altitude of the frustum)
A₁ = area of the lower base
A₂ = area of the upper base
Note that A₁ and A₂ are parallel to each other.

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Problem 341
Derive the torsion formula τ = Tρ / J for a solid circular section by assuming the section is composed of a series of concentric thin circular tubes. Assume that the shearing stress at any point is proportional to its radial distance.

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The Double Angle Formulas can be derived from Sum of Two Angles listed below:
$\sin (A + B) = \sin A \, \cos B + \cos A \, \sin B$ → Equation (1)

$\cos (A + B) = \cos A \, \cos B - \sin A \, \sin B$ → Equation (2)

$\tan (A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \, \tan B}$ → Equation (3)

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For two numbers x and y, let x, a, y be a sequence of three numbers. If x, a, y is an arithmetic progression then 'a' is called arithmetic mean. If x, a, y is a geometric progression then 'a' is called geometric mean. If x, a, y form a harmonic progression then 'a' is called harmonic mean.

Let AM = arithmetic mean, GM = geometric mean, and HM = harmonic mean. The relationship between the three is given by the formula

$AM \times HM = GM^2$

Below is the derivation of this relationship.