Mathmatic Equations using in HTML5 Radio

Mathmatic Equations used in HTML5 Radio

y^{'} = f^{'} (x) = \frac{d y}{d x} = lim_{h \to 0} \frac{f (x + h) - f (h)}{h}

\cos^{2} θ + \sin^{2} θ = 1

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2 R

a^{2} = b^{2} + c^{2} - 2 b c \cos A

\log_{a} x y = \log_{a} x + \log_{a} y

\log_{a} x = \frac{\log_{b} x}{\log_{b} a}

\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}

\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}

\sum_{k = 1}^{n} k^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}

\int x^{a} d x = \frac{x^{a + 1}}{a + 1} + C (a \neq - 1)

{(a+b)}^{2} = a^{2} + 2ab + b^{2}

\int \frac{1}{x} = \log |x| + C

\lim_{x \to 0} \frac{\sin x}{x} = 1

\lim_{x \to 0} {(1 + x)}^{\frac{1}{x}} = e

\lim_{x \to 0} \frac{e^{x} - 1}{x} = 1

\frac{a_{1} + a_{2}}{2} \geq \sqrt{a_{1} a_{2}}

\frac{a_{1} + a_{2} + a_{3}}{3} \geq \sqrt[3]{{a_{1} a}_{2} a_{3}}

({x_{1}}^{2} + {x_{2}}^{2}) ({y_{1}}^{2} + {y_{2}}^{2}) \geq {(x_{1} y_{1} + x_{2} y_{2})}^{2}

{(\cos θ + i \sin θ)}^{n} = \cos n θ + i \sin n θ

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos θ

\vec{a} \cdot \vec{b} = a_{1} b_{1} + a_{2} b_{2}

(a + b) (a - b) = a^{2} - b^{2}

a^{2} + b^{2} + c^{2} - a b - b c - c a \geq 0

\int f (x) g^{'} (x) d x = f (x) g (x) - \int f^{'} (x) g (x) d x

\int f (x) = \int f (φ (t)) φ^{'} (t) d t = \int f (x) \frac{d x}{d t} d t

\int \frac{1}{1 + x^{2}} = \tan^{-1} x

\int \frac{1}{\sqrt{1 - x^{2}}} d x = \sin^{-1} x or - \cos^{-1} x

\int_{- a}^{a} \sqrt{a^{2} - x^{2}} d x = \frac{π a^{2}}{2}

\int_{0}^{1} f (x) d x = \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{1}{n} f (\frac{k}{n})

\bar{(A \cap B)} = \bar{A} \cup \bar{B}

\bar{(A \cup B)} = \bar{A} \cap \bar{B}

\sqrt{(a + b) \pm 2 \sqrt{a b}} = \sqrt{a} \pm \sqrt{b}

{(a + b)}^{n} = \sum_{k = 0}^{n}_{n} C_{k} a^{n - k} b^{k}

(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} e & f \\ g & h \end{matrix}) = (\begin{matrix} a e + b g & a f + b h \\ c e + d g & c f + d h \end{matrix})

A^{-1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix})

For A = (\begin{matrix} a & b \\ c & d \end{matrix}), A^{2} - (a + d) A + (a d - b c) E = O

y = f (x) g (x) \Rightarrow y^{'} = f^{'} (x) g (x) + f (x) g^{'} (x)

y = \frac{g (x)}{f (x)} \Rightarrow y^{'} = \frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{{\{f (x)\}}^{2}}

y = \log (|x|) \Rightarrow y^{'} = \frac{1}{x}

y = e^{x} \Rightarrow y^{'} = e^{x}

y = x^{n} \Rightarrow y^{'} = n x^{n - 1}

y = \sin x \Rightarrow y^{'} = \cos x

y = \cos x \Rightarrow y^{'} = - \sin x

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

y = \tan x \Rightarrow y^{'} = \frac{1}{\cos^{2} x}

e^{i θ} = \cos (θ) + i \sin (θ)

n! \sim \sqrt{2 π n} {(\frac{n}{e})}^{n}

\sin (α + β) = \sin (α) \cos (β) + \cos (α) \sin (β)

\cos (α + β) = \cos (α) \cos (β) - \sin (α) \sin (β)

\tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β}