Maths Formula Compendium

Trigonometry

Name	$0\degree$	$30\degree (\frac{\pi}{6})$	$45\degree (\frac{\pi}{4})$	$60\degree (\frac{\pi}{3})$	$90\degree (\frac{\pi}{2})$	$120\degree (\frac{2\pi}{3})$	$135\degree (\frac{3\pi}{4})$	$150\degree (\frac{5\pi}{6})$	$180\degree (\pi)$
$\sin$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\cos$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-\frac{1}{\sqrt{2}}$	$-\frac{\sqrt{3}}{2}$	$-1$
$\tan$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	$\infty$	$-\sqrt{3}$	$-1$	$-\frac{1}{\sqrt{3}}$	$0$
$\cot$	$\infty$	$\sqrt{3}$	$1$	$\frac{1}{\sqrt{3}}$	$0$	$-\frac{1}{\sqrt{3}}$	$-1$	$-\sqrt{3}$	$-\infty$
$\sec$	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$ 2 $	$\infty$	$-2$	$-\sqrt{2}$	$-\frac{2}{\sqrt{3}}$	$-1$
$\cosec$	$\infty$	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	$\infty$

Trigonometric Identities

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

$\cos(A - B) = \cos A \cos B + \sin A \sin B$

$\sin C + \sin D = 2 \sin \frac{C + D}{2} \cos \frac{C - D}{2}$

$\sin C - \sin D = 2 \cos \frac{C + D}{2} \sin \frac{C - D}{2}$

$\cos C + \cos D = 2 \cos \frac{C + D}{2} \cos \frac{C - D}{2}$

$\cos C - \cos D = -2 \sin \frac{C + D}{2} \sin \frac{C - D}{2}$

$\sin(A + B) \sin(A - B) = \sin^2 A - \sin^2 B$

$\cos(A + B) \cos(A - B) = \cos^2 A - \sin^2 B$

$\tan A - \tan B = \frac{\sin(A - B)}{\cos A \cos B}$

$\cot A - \cot B = \frac{-\sin(A - B)}{\sin A \sin B}$

\[ \sin(A + B + C) = \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C \]

\[ \cos(A + B + C) = \cos A \cos B \cos C - \cos A \sin B \sin C - \sin A \cos B \sin C - \sin A \sin B \cos C \]

\[ \tan(A + B + C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1 - \tan A \tan B - \tan B \tan C - \tan C \tan A} \]

Double Angle Formulas

$$ \begin{aligned} \sin 2A &= 2 \sin A \cos A = \frac{2 \tan A}{1 + \tan^2 A} \\ \cos 2A &= \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A = \frac{1 - \tan^2 A}{1 + \tan^2 A} \end{aligned} $$

$$ \begin{aligned} \sin 3A &= 3 \sin A - 4 \sin^3 A \\ \cos 3A &= 4 \cos^3 A - 3 \cos A \end{aligned} $$

$$ \begin{aligned} \tan 2A &= \frac{2 \tan A}{1 - \tan^2 A} \\ \tan 3A &= \frac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A} \end{aligned} $$

Product-to-Sum Formulas

$2 \sin x \sin y = \cos(x - y) - \cos(x + y)$
$2 \cos x \cos y = \cos(x + y) + \cos(x - y)$
$2 \sin x \cos y = \sin(x + y) + \sin(x - y)$
$2 \cos x \sin y = \sin(x + y) - \sin(x - y)$

Pythagorean Identities

$\sin^2 \theta + \cos^2 \theta = 1$
$1 + \tan^2 \theta = \sec^2 \theta$
$1 + \cot^2 \theta = \csc^2 \theta$

$\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$

$\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$

$\cot(x + y) = \frac{\cot x \cot y - 1}{\cot x + \cot y}$

$\cot(x - y) = \frac{\cot x \cot y + 1}{\cot y - \cot x}$

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

General Solutions

Basic Cases

$\sin \theta = 0 \implies \theta = n\pi$
$\cos \theta = 0 \implies \theta = (2n + 1)\frac{\pi}{2}$
$\tan \theta = 0 \implies \theta = n\pi$

Equality Cases

$\sin \theta = \sin \beta \implies \theta = n\pi + (-1)^n \beta$
$\cos \theta = \cos \beta \implies \theta = 2n\pi \pm \beta$
$\tan \theta = \tan \beta \implies \theta = n\pi + \beta$

Derived Cases

$\sin^2 \theta = \sin^2 \beta \implies \theta = n\pi \pm \beta$
$\cos^2 \theta = \cos^2 \beta \implies \theta = n\pi \pm \beta$
$\tan^2 \theta = \tan^2 \beta \implies \theta = n\pi \pm \beta$

Secant and Cosine Properties

$-\cos \theta = \cos(\pi - \theta)$
$-\sec \theta = \sec(\pi - \theta)$
$\cos(- \theta) = \cos\theta$
$\sec(-\theta) = \sec\theta$

Transformation Rule (TR)

$(\frac{n\pi}{2} + \theta)$

If $n$ is odd $\Rightarrow$ TR will change.

$(n\pi + \theta)$

If $n$ is odd/even $\Rightarrow$ TR will not change.

Inverse Trigonometry

	Domain (value) ($x$)	Range (Angle) ($\theta$)
$\sin^{-1}x$	$[-1,1]$	$[-\pi/2, \pi/2]$
$\cos^{-1}x$	$[-1,1]$	$[0, \pi]$
$\tan^{-1}x$	$\mathbb{R}$	$(-\pi/2, \pi/2)$
$\cot^{-1}x$	$\mathbb{R}$	$(0, \pi)$
$\sec^{-1}x$	$(-\infty,-1] \cup [1,\infty)$	$[0,\pi] \setminus \{\pi/2\}$
$\cosec^{-1}x$	$(-\infty,-1] \cup [1,\infty)$	$[-\pi/2,0) \cup (0,\pi/2]$

Property 1	Property 2
$\sin^{-1}(\sin\theta) = \theta$	$\sin(\sin^{-1}x) = x$
$\cos^{-1}(\cos\theta) = \theta$	$\cos(\cos^{-1}x) = x$
$\tan^{-1}(\tan\theta) = \theta$	$\tan(\tan^{-1}x) = x$
$\cosec^{-1}(\cosec\theta) = \theta$	$\cosec(\cosec^{-1}x) = x$
$\sec^{-1}(\sec\theta) = \theta$	$\sec(\sec^{-1}x) = x$
$\cot^{-1}(\cot\theta) = \theta$	$\cot(\cot^{-1}x) = x$

These compositions use the principal ranges in the table: for example, $\sin^{-1}(\sin\theta)=\theta$ only when $\theta\in[-\pi/2,\pi/2]$, and $\cos^{-1}(\cos\theta)=\theta$ only when $\theta\in[0,\pi]$.

Property 3
$\sin^{-1}(-x) = -\sin^{-1}(x)$	$\cos^{-1}(-x) = \pi -\cos^{-1}(x)$
$\tan^{-1}(-x) = -\tan^{-1}(x)$	$\cot^{-1}(-x) = \pi -\cot^{-1}(x)$
$\cosec^{-1}(-x) = -\cosec^{-1}(x)$	$\sec^{-1}(-x) = \pi -\sec^{-1}(x)$

Property 4
$\sin^{-1}(\frac{1}{x}) = \cosec^{-1}(x)$
$\cos^{-1}(\frac{1}{x}) = \sec^{-1}(x)$
$\tan^{-1}(\frac{1}{x}) = \cot^{-1}(x),\quad x>0$

The reciprocal identities for inverse sine and inverse cosine require $|x|\geq1$.

For $x<0$, $\tan^{-1}(1/x)=\cot^{-1}(x)-\pi$ with the stated principal range for $\cot^{-1}$.

Property 5
$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$
$\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$
$\sec^{-1}x + \cosec^{-1}x = \frac{\pi}{2}$

Property 6
$\sin^{-1}x + \sin^{-1}y = \sin^{-1} \big(x\sqrt{1-y^2} + y\sqrt{1-x^2}\big)$
$\sin^{-1}x - \sin^{-1}y = \sin^{-1} \big(x\sqrt{1-y^2} - y\sqrt{1-x^2}\big)$

Property 7
$\cos^{-1}x + \cos^{-1}y = \cos^{-1} \big(xy - \sqrt{1-x^2} \sqrt{1-y^2}\big)$
$\cos^{-1}x - \cos^{-1}y = \cos^{-1} \big(xy + \sqrt{1-x^2} \sqrt{1-y^2}\big)$

For Properties 6 and 7, use these identities only when the resulting angle is in the principal inverse-trigonometric range; otherwise apply the appropriate branch adjustment.

Property 8
$\tan^{-1}x - \tan^{-1}y \equiv \tan^{-1}\Big(\frac{x-y}{1+xy}\Big) \pmod{\pi}$
$\tan^{-1}x + \tan^{-1}y \equiv \tan^{-1}\Big(\frac{x+y}{1-xy}\Big) \pmod{\pi}$

These sum-and-difference identities assume the denominators are nonzero. Inverse-trigonometric results use the principal ranges above; adjust by a multiple of $\pi$ when needed.

Property 9

$\sin\big(2\sin^{-1}x\big) = 2x\sqrt{1-x^2}$

$\sin\big(3\sin^{-1}x\big) = 3x-4x^3$

$\cos\big(2\cos^{-1}x\big) = 2x^2-1$

$\cos\big(3\cos^{-1}x\big) = 4x^3-3x$

$$ \begin{aligned} 2\tan^{-1}x &\equiv \tan^{-1}\Big(\frac{2x}{1-x^2}\Big) \pmod{\pi} \\ 3\tan^{-1}x &\equiv \tan^{-1}\Big(\frac{3x-x^3}{1-3x^2}\Big) \pmod{\pi} \end{aligned} $$

Property 10

\[ \begin{aligned} \sin\big(2\tan^{-1}x\big) &= \frac{2x}{1+x^2} \\ \cos\big(2\tan^{-1}x\big) &= \frac{1-x^2}{1+x^2} \end{aligned} \]

$1-\cos\theta = 2 \sin^2(\theta/2)$

$1+ \cos\theta = 2\,\cos^2(\theta/2)$

$\sin\theta = 2\sin(\theta/2)\cos(\theta/2)$

$2\sin^{-1}x + \sin^{-1}(-x) = \cos^{-1}x$

$$ \begin{split}\cos2\theta &= \cos^2\theta - \sin^2\theta \\& = 2\cos^2\theta -1 \\ &= 1 - 2\sin^2\theta \end{split} $$

Expression and Substitution

Expression	Substitution	Substitution
$a^2 + x^2$	$x = a\tan\theta$	$x = a\cot\theta$
$a^2-x^2$	$x=a\sin\theta$	$x=a\cos\theta$
$x^2-a^2$	$x=a\sec\theta$	$x=a\cosec\theta$
$\sqrt{\frac{a-x}{a+x}}$	$x=a\cos2\theta$
$\sqrt{\frac{a^2+x^2}{a^2-x^2}}$	$x=a^2\cos2\theta$

Differentiation

$\frac{d}{dx} x^n = n x^{n-1}$

$\frac{d}{dx} a^x = a^x \ln a$

$\frac{d}{dx} \log_e x = \frac{1}{x}$

$\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$

$\frac{d}{dx} e^x = e^x$

$\frac{d}{dx} \sin x = \cos x$

$\frac{d}{dx} \cos x = -\sin x$

$\frac{d}{dx} \tan x = \sec^2 x$

$\frac{d}{dx} \sec x = \sec x \tan x$

$\frac{d}{dx} \cot x = -\csc^2 x$

$\frac{d}{dx} \cosec x = -\cosec x \cot x$

$\log m^n = n \log m$

$\log m + \log n = \log (mn)$

$\log m - \log n = \log \left(\frac{m}{n}\right)$

$\log e = 1$, $\log_a a = 1$

$\log_a b = \frac{1}{\log_b a}$

$\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}$

$\frac{d}{dx} \cos^{-1} x = \frac{-1}{\sqrt{1 - x^2}}$

$\frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2}$

$\frac{d}{dx} \cot^{-1} x = \frac{-1}{1 + x^2}$

$\frac{d}{dx} \sec^{-1} x = \frac{1}{x \sqrt{x^2 - 1}}$

$\frac{d}{dx} \cosec^{-1} x = \frac{-1}{x \sqrt{x^2 - 1}}$

$a^{\log_a x} = x$
$x^{\log_a y} = y^{\log_a x}$

Chain Rule

If a variable $z$ depends on the variable $y$, which itself depends on the variable $x$ (that is, $y$ and $z$ are dependent variables), then $z$ depends on $x$ as well, via the intermediate variable $y$. In this case, the chain rule is expressed as:

$$ \frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx} $$

Integration Formulas

Basic Integration Rules

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$
$\int \frac{1}{x} \, dx = \ln |x| + C$
$\int e^x \, dx = e^x + C$
$\int a^x \, dx = \frac{a^x}{\ln a} + C$
$\int \sin x \, dx = -\cos x + C$
$\int \cos x \, dx = \sin x + C$
$\int \sec^2 x \, dx = \tan x + C$
$\int \csc^2 x \, dx = -\cot x + C$
$\int \sec x \tan x \, dx = \sec x + C$
$\int \csc x \cot x \, dx = -\csc x + C$
$\int \cot x \, dx = \ln |\sin x| + C$
$\int \tan x \, dx = -\ln |\cos x| + C = \ln |\sec x| + C$
$\int \sec x \, dx = \ln |\sec x + \tan x| + C$
$\int \csc x \, dx = \ln |\csc x - \cot x| + C$

Inverse Trigonometric Integrals

$\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1} \left(\frac{x}{a}\right) + C$
$\int \frac{-1}{\sqrt{a^2 - x^2}} \, dx = \cos^{-1} \left(\frac{x}{a}\right) + C$
$\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) + C$
$\int \frac{-1}{x^2 + a^2} \, dx = \frac{1}{a} \cot^{-1} \left(\frac{x}{a}\right) + C$
$\int \frac{1}{x \sqrt{x^2 - a^2}} \, dx = \frac{1}{a} \sec^{-1} \left|\frac{x}{a}\right| + C$
$\int \frac{-1}{x \sqrt{x^2 - a^2}} \, dx = \frac{1}{a} \csc^{-1} \left|\frac{x}{a}\right| + C$

Logarithmic and Advanced Integrals

$\int e^{x} \big(f(x) + f'(x)\big) \, dx = e^x f(x) + C$
$\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln |x + \sqrt{x^2 + a^2}| + C$
$\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln \left|\frac{x - a}{x + a}\right| + C$
$\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left|\frac{a + x}{a - x}\right| + C$
$\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left|\frac{a + x}{x - a}\right| + C$

Additional Properties

$\int k f(x) \, dx = k \int f(x) \, dx$
$\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$
$\int c \, dx = c x + C$

Line Equations

$ax + by + c=0$

$m = -a/b$ (Slope of line)
One Point form of line

\[ y - y_1 = m (x-x_1) \]
Two point form of line

$$ y-y_1 = \frac{y_2-y_1}{x_2-x_1}(x-x_1) $$

Intercept Form of line

$$ \frac{x}{a} + \frac{y}{b} = 1 $$

Normal form of line

$$ x\cos \theta + y\sin\theta = P $$

Point Slope form

$$ \begin{aligned} y &= mx+c \ m &= \frac{y_2-y_1}{x_2-x_1} \end{aligned} $$

Distance of a point from a line

\[ \text{Dist}_{PA}= \Bigg|{\frac{ax_1 + by_1 + c}{\sqrt{a^2+b^2}}}\Bigg| \]

Distance between two lines

$y = mx + c_1 \qquad y= mx+c_2$

\[ \text{d} = \Bigg| \frac{c_1 - c_2}{\sqrt{1+m^2}}\Bigg| = \Bigg|\frac{c_1 - c_2}{\sqrt{a^2+b^2}}\Bigg| \]

Angle between two lines

Where $m_1$ and $m_2$ are slopes of two lines.

\[ \tan\theta = \frac{m_2 - m_1}{1+m_2m_1} \]

If line $l_1$ and $l_2$ are orthogonal to each other, then. $m_1m_2 = -1$
Collinearity of points

Slope of $AB$ = Slope of $AC$

Shapes CSA(Curved Surface Area), T(Total)SA, and volume

Frustum

$\text{CSA} = \pi l (r_1+ r_2)$

$\text{TSA} = \pi r_1^2 + \pi r_2^2 + \pi l(r_1+r_2)$

$\text{Volume} = \frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1r_2)$

where $l= \sqrt{h^2+(r_1-r_2)^2}$

Adjoint and Inverse

$A(\text{adj} A) = \|A\| I_n = (\text{adj}) A$

$A^{-1} = \frac{1}{\|A\|} (\text{adj} A)$

$(A^\top)^{-1} = (A^{-1})^\top$

$\|A \enspace adj A\| = \|A\|^{n}$

$\|A^\top\| = \|A\| $

$AA^{-1} = I_n$

$(A^{-1})^{-1} = A$

$(AB)^{-1} = B^{-1}A^{-1}$

$\text{adj}\space AB = (\text{adj} B)(\text{adj} A)$

$\|AB\| = \|A\| \|B\|$

$\text{adj}A^\top = (\text{adj} A)\top$

$\|KA\| = K^n \|A\|$

$AA^{-1}=I$

$A^{-1}I = A^{-1}$

$\|\text{adj} A\| = \|A\|^{n-1}$

$\text{adj} \space(adj A) = \|A\|^{n-2} A$

$\|\text{adj} \space(adj A)\| = \|A\|^{(n-1)^{2}}$

Binomial Theorem

Initial conditions:

$^nc_0 = 1$

$^nc_1 = n$

$^nc_n = 1$

$^nc_{n-1} = n$

Basic expansion:

\[(x+a)^n = {^nc_0}x^na^0 + {^nc_1}x^{n-1}a^1 + {^nc_2}x^{n-2}a^2 + ... + {^nc_n}x^0a^n\]

Sum and difference formulas:

\[ (x+a)^n + (x-a)^n = 2\left[{^nc_0}x^na^0 + {^nc_2}x^{n-2}a^2 + {^nc_4}x^{n-4}a^4 + \cdots\right] \]

\[ (x+a)^n - (x-a)^n = 2\left[{^nc_1}x^{n-1}a^1 + {^nc_3}x^{n-3}a^3 + {^nc_5}x^{n-5}a^5 + \cdots\right] \]

Number of terms:

	When n is odd	When n is even
(x+a)^n + (x-a)^n	$(\frac{n+1}{2})$ terms	$(\frac{n}{2})$ terms
(x+a)^n - (x-a)^n	$(\frac{n+1}{2})$ terms	$(\frac{n}{2})$ terms

General Term and Middle Term

General term: $t_{r+1} = {^nc_r}x^{n-r}a^r$

Middle term occurs at:

If n is odd: $(\frac{n+1}{2})$ & $(\frac{n+3}{2})$ terms
If n is even: $(\frac{n}{2}+1)$ terms

Coefficient Tables

Coefficient of	Binomial Expression	is
$(r+1)^{\text{th}}$	$(1+x)^n$	${^nc_r}$
$x^r$	$(1+x)^n$	${^nc_r}$
$x^r$	$(1-x)^n$	$(-1)^r\enspace{^nc_r}$
$(r+1)^{\text{th}}$	$(1-x)^n$	$(-1)^r\enspace{^nc_r}$

Relations

$\frac{^nc_r}{^nc_{r-1}} = \frac{n-r+1}{r}$

$^nc_r = (\frac{n}{r})\enspace{^{n-1}c_{r-1}}$

$\frac{^nc_{r+1}}{^nc_r} = \frac{n-r}{r+1}$

$\frac{t_{r+1}}{t_r} = \frac{n-r+1}{r} \cdot \frac{a}{x}$

Probability

$P_r = \frac{n!}{(n-r)!}$

$C_r = \frac{n!}{(n-r)!r!}$

Operation	Notation
A or B	$A \cup B$
A and B	$A \cap B$
A but not B	$A \cap \bar{B}$
B but not A	$\bar{A} \cap B$
Neither A nor B	$\bar{A} \cap \bar{B}$
At least one of A,B,& C	$A \cup B \cup C$
Exactly one of A,B	$(A \cap \bar{B}) \cup (\bar{A} \cap B)$
All Three of A,B,& C	$A \cap B \cap C$
Exactly Two of A,B,& C	$(A \cap B \cap \bar{C}) \cup (A \cap \bar{B} \cap C) \cup (\bar{A} \cap B \cap C)$
Exactly One of A,B,& C	$(A \cap \bar{B} \cap \bar{C}) \cup (\bar{A} \cap B \cap \bar{C}) \cup (\bar{A} \cap \bar{B} \cap C)$

The probability formulas:

$$ \begin{split} P(A \cup B) &= P(A) + P(B) - P(A \cap B) \\ &= 1 - P(\overline{A \cup B}) \\ &= 1 - P(\overline{A} \cap \overline{B}) \\ &= 1 - P(\bar{A})P(\bar{B}) \end{split} $$

$$ \tag{\text{A and B are mutually exclusive}} P(A \cup B) = P(A) + P(B) $$

$$ \begin{split} P(A \cup B \cup C) &= P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C) \end{split} $$

$$ \tag{\text{A, B, and C are mutually exclusive}} P(A \cup B \cup C) = P(A) + P(B) + P(C) $$

Probability of occurrence of (A) only $$P(A \cap \bar{B}) = P(A) - P(A \cap B)$$
Probability of occurrence of (B) only $$P(\bar{A} \cap B) = P(B) - P(A \cap B)$$
P of occurrence of exactly one of A and B $$P(A \cap \bar{B}) + P(\bar{A} \cap B) = P(A \cup B) - P(A \cap B)$$
Three Events Occurring Simultaneously

$$P(A \cap B \cap C)$$

At Least Two Events: $$P(A \cap B) + P(B \cap C) + P(C \cap A) - 2P(A \cap B \cap C)$$
Exactly Two Out of Three Events: $$P(A \cap B) + P(B \cap C) + P(C \cap A) - 3P(A \cap B \cap C)$$
Exactly One Out of Three Events: $$P(A) + P(B) + P(C) - 2P(A \cap B) - 2P(B \cap C) - 2P(C \cap A) + 3P(A \cap B \cap C)$$
Exactly One Out of Two Events $$P(A \cap \overline{B}) + P(\overline{A} \cap B)$$
- Simplified Form: $$P(A) + P(B) - 2P(A \cap B) \implies P(A \cup B) - P(A \cap B)$$
Conditional probability P of E given A is true

$$ \begin{split} P(E_i|A) &= P(E_i)P(A|E_i) \\ &= \sum_{i=1}^n P(E_i)P(A|E_i) \end{split} $$

$P(A \cap B) = P(A)P(B|A) = P(B)P(A|B)$

$P(A \cap B \cap C \cap D) = P(A)P(B|A)P(C|A \cap B)P(D|A \cap B \cap C)$

\[P(E_i | A) = \frac{P(E_i) P(A|E_i)}{\sum^n_{i=1}P(E_i) P(A|E_i)}\]

Also noting the card suit groupings shown in the image:

Hearts and Diamonds are grouped as Red
Spades and Clubs are grouped as Black

Mean Variance & Standard Deviation

$p$ means success event
$q$ means not a sucess event

$P(X = r) = {}_nC_r p^r q^{n-r}$

$Variance = npq$

$Mean = np$

Probability Relations: $$P(X \leq x_i) = P(X = x_1) + P(X = x_2) + ... + P(X = x_i) = p_1 + p_2 + ... + p_i$$

\[P(X < x_i) = P(X = x_1) + P(X = x_2) + ... + P(X = x_{i-1}) = p_1 + p_2 + ... + p_{i-1}\]

\[P(X \geq x_i) = P(X = x_i) + P(X = x_{i+1}) + ... + P(X = x_n) = p_i + p_{i+1} + ... + p_n\]

\[P(X > x_i) = P(X = x_{i+1}) + P(X = x_{i+2}) + ... + P(X = x_n) = p_{i+1} + p_{i+2} + ... + p_n\]

$P(X \geq x_i) = 1 - P(X < x_i)$

$P(X \leq x_i) = 1 - P(X > x_i)$

$P(X > x_i) = 1 - P(X \leq x_i)$

$P(X < x_i) = 1 - P(X \geq x_i)$

\[P(x_i \leq X \leq x_j) = P(X = x_i) + P(X = x_{i+1}) + ... + P(X = x_j)\]

\[P(x_i < X < x_j) = P(X = x_{i+1}) + P(X = x_{i+2}) + ... + P(X = x_{j-1})\]

Mean Formulas(Matehmatical expectation):

where $f$ = frequency

$$ \begin{split} \overline{X} &= p_1x_1 + p_2x_2 + ... + p_nx_n \\ &= E(X) \\ &= \sum_{i=1}^n p_ix_i \\ &= \frac{f_1x_1}{N} + \frac{f_2x_2}{N} + ... + \frac{f_nx_n}{N} \end{split} $$

\[p_i = \frac{f_i}{N}\]

Variance Formula:

$$ \begin{split} Var(X) &= E(X^2) - \lbrace E(X)\rbrace^2 \\ &= \sum_{i=1}^n p_ix_i^2 - (\sum_{i=1}^n p_ix_i)^2 \end{split} $$

\[\frac{\sum x^2}{n} - (\frac{\sum x}{n})^2\]

Random Variable Property:

If $X$ has mean $E(X)$ and variance $Var(X)$, then $aX + b$ has mean $aE(X) + b$ and variance $a^2Var(X)$.

Standard Deviation:

$$ \begin{aligned} \sigma &= \sqrt{Var(X)} \\ \sigma^2 &= \frac{1}{N^2} \left[N\sum_{i=1}^n f_i x_i^2 - \left(\sum_{i=1}^n f_i x_i\right)^2\right] \end{aligned} $$

Arithmetic Progression (AP)

General Form: $A, B, C$

$$2B = A + C$$

Formulas:

$t_n = a + (n-1)d$
$S_n = \frac{n}{2} \left[2a + (n-1)d \right]$
$S_n = \frac{n}{2} [a + a_n]$
$t_n = S_n - S_{n-1}$

Arithmetic Mean (AM):

$G = \frac{a+b}{2}$

Geometric Progression (GP)

General Form: $A, B, C$

$$B^2 = A \cdot C$$

Formulas:

$t_n = a \cdot r^{n-1}$
$S_n = a \frac{r^n - 1}{r - 1}, \, r \neq 1$
$S_\infty = \frac{a}{1-r}, \, |r| < 1$
where $r$ is common ratio

Geometric Mean (GM):

Single Mean Between $a$ and $b$: $G = \sqrt{ab}$
Multiple Means:
$G_1 = ar^1$
$G_2 = ar^2$
$G_3 = ar^3$
$r = \sqrt[n+1]{\frac{b}{a}}$

AP, GP, HP Relation:

\[AP \geq GP \geq HP\]

Special Sequences:

Sum of Natural Numbers:

$$ 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} $$

Sum of Squares of Natural Numbers:

$$ 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6} $$

Sum of Cubes of Natural Numbers:

$$ 1^3 + 2^3 + 3^3 + \dots + n^3 = \left[\frac{n(n+1)}{2}\right]^2 $$

Homogeneous and Non homogeneous equations solutions

NON-HOMOGENEOUS

$AX = B$

Unique Solution

$\text{adj}(A)B = 0 \implies \text{Infinite Solutions}$
$\text{adj}(A)B \neq 0 \implies \text{No Solution}$

When $|A| \neq 0$

When $|A| = 0$

HOMOGENEOUS

$AX = 0$

Trivial solution $x = y = z = 0 \enspace $:

$\text{Infinite Solutions}$ put $z = k$ and solve

Finding Log

Given we need to find $\log$ of $\log 15.27$
- Move the decimal after 1st digit and introduce power of 10.
\[ \log \textcolor{#f56c42}1.\textcolor{#f56c42}5\textcolor{#ad42f5}2\textcolor{#f542bc}7 \times 10^{\textcolor{#42f5f2}1} \]

When the decimal is moved in left/right: $$ (-)\medspace \overrightarrow{\text{introduce negative powers}} \qquad \overleftarrow{\text{introduce positive powers}} \medspace(+) $$
Look for $\textcolor{#f56c42}{15}$^th row and column with label $\textcolor{#ad42f5}2$. which is $\bold{\textcolor{#07fc03}{1818}}$.
Add Mean difference from column $\textcolor{#f542bc}7$ in the corresponding row. which is $\textcolor{#07fc03}{20}$

\[ 1818 + 20 = \bold{\textcolor{#07fc03}{1838}} \]
Write the exponent, insert decimal and write the value calculated in Step 3.

\[ \textcolor{#42f5f2}1.\textcolor{#07fc03}{1838} \]
So $\log 15.27 = \bold{\textcolor{#07fc03}{1.1838}}$

Finding AntiLog

Given we need to find Antilog of $15.5932$ $$ \begin{aligned} \log k &= 15.5932 \ k &= \operatorname{Antilog}(\medspace \textcolor{#42f5f2}{15}\textcolor{#f56c42}{.59}\textcolor{#ad42f5}3\textcolor{#f542bc}2\medspace) \end{aligned} $$
Look for $0\textcolor{#f56c42}{.59}$^th row and column with label $\textcolor{#ad42f5}3$. Which is $\bold{\textcolor{#07fc03}{3917}}$.
Add Mean difference from column $\textcolor{#f542bc}2$ to previous result. Which is $\textcolor{#07fc03}2$

\[ 3917 + 2 = \bold{\textcolor{#07fc03}{3919}} \]
Add $1$ to characteristic $\textcolor{#42f5f2}{15} = \textcolor{#07fc03}{16}$ and add insert the decimal from left calculated in step 2. That means we need to add decimal after $\textcolor{#07fc03}{16}$^th position

\[ \begin{aligned} 3919\enspace0000\enspace0000\enspace000 .\\ \implies k = \bold{\textcolor{#07fc03}{3.919\times10^{15}}} \end{aligned} \]

Quickly write the exponential form

Since, we want to put decimal just after 1^st digit. We need to move decimal from 16^th position to right after 1^st digit; which will introduce +ve powers. ($16 -1 = \textcolor{#07fc03}{15}$). Since we moved 15 positions left.

$3.919 \times 10^{\textcolor{#07fc03}{15}}$

Name	\(0\degree\)	\(30\degree (\frac{\pi}{6})\)	\(45\degree (\frac{\pi}{4})\)	\(60\degree (\frac{\pi}{3})\)	\(90\degree (\frac{\pi}{2})\)	\(120\degree (\frac{2\pi}{3})\)	\(135\degree (\frac{3\pi}{4})\)	\(150\degree (\frac{5\pi}{6})\)	\(180\degree (\pi)\)
\(\sin\)	\(0\)	\(\frac{1}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{\sqrt{3}}{2}\)	\(1\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{1}{2}\)	\(0\)
\(\cos\)	\(1\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{1}{2}\)	\(0\)	\(-\frac{1}{2}\)	\(-\frac{1}{\sqrt{2}}\)	\(-\frac{\sqrt{3}}{2}\)	\(-1\)
\(\tan\)	\(0\)	\(\frac{1}{\sqrt{3}}\)	\(1\)	\(\sqrt{3}\)	\(\infty\)	\(-\sqrt{3}\)	\(-1\)	\(-\frac{1}{\sqrt{3}}\)	\(0\)
\(\cot\)	\(\infty\)	\(\sqrt{3}\)	\(1\)	\(\frac{1}{\sqrt{3}}\)	\(0\)	\(-\frac{1}{\sqrt{3}}\)	\(-1\)	\(-\sqrt{3}\)	\(-\infty\)
\(\sec\)	\(1\)	\(\frac{2}{\sqrt{3}}\)	\(\sqrt{2}\)	$ 2 $	\(\infty\)	\(-2\)	\(-\sqrt{2}\)	\(-\frac{2}{\sqrt{3}}\)	\(-1\)
\(\cosec\)	\(\infty\)	\(2\)	\(\sqrt{2}\)	\(\frac{2}{\sqrt{3}}\)	\(1\)	\(\frac{2}{\sqrt{3}}\)	\(\sqrt{2}\)	\(2\)	\(\infty\)

	Domain (value) (\(x\))	Range (Angle) (\(\theta\))
\(\sin^{-1}x\)	\([-1,1]\)	\([-\pi/2, \pi/2]\)
\(\cos^{-1}x\)	\([-1,1]\)	\([0, \pi]\)
\(\tan^{-1}x\)	\(\mathbb{R}\)	\((-\pi/2, \pi/2)\)
\(\cot^{-1}x\)	\(\mathbb{R}\)	\((0, \pi)\)
\(\sec^{-1}x\)	\((-\infty,-1] \cup [1,\infty)\)	\([0,\pi] \setminus \{\pi/2\}\)
\(\cosec^{-1}x\)	\((-\infty,-1] \cup [1,\infty)\)	\([-\pi/2,0) \cup (0,\pi/2]\)

Property 1	Property 2
\(\sin^{-1}(\sin\theta) = \theta\)	\(\sin(\sin^{-1}x) = x\)
\(\cos^{-1}(\cos\theta) = \theta\)	\(\cos(\cos^{-1}x) = x\)
\(\tan^{-1}(\tan\theta) = \theta\)	\(\tan(\tan^{-1}x) = x\)
\(\cosec^{-1}(\cosec\theta) = \theta\)	\(\cosec(\cosec^{-1}x) = x\)
\(\sec^{-1}(\sec\theta) = \theta\)	\(\sec(\sec^{-1}x) = x\)
\(\cot^{-1}(\cot\theta) = \theta\)	\(\cot(\cot^{-1}x) = x\)

Property 3
\(\sin^{-1}(-x) = -\sin^{-1}(x)\)	\(\cos^{-1}(-x) = \pi -\cos^{-1}(x)\)
\(\tan^{-1}(-x) = -\tan^{-1}(x)\)	\(\cot^{-1}(-x) = \pi -\cot^{-1}(x)\)
\(\cosec^{-1}(-x) = -\cosec^{-1}(x)\)	\(\sec^{-1}(-x) = \pi -\sec^{-1}(x)\)

Expression	Substitution	Substitution
\(a^2 + x^2\)	\(x = a\tan\theta\)	\(x = a\cot\theta\)
\(a^2-x^2\)	\(x=a\sin\theta\)	\(x=a\cos\theta\)
\(x^2-a^2\)	\(x=a\sec\theta\)	\(x=a\cosec\theta\)
\(\sqrt{\frac{a-x}{a+x}}\)	\(x=a\cos2\theta\)
\(\sqrt{\frac{a^2+x^2}{a^2-x^2}}\)	\(x=a^2\cos2\theta\)

	When n is odd	When n is even
(x+a)^n + (x-a)^n	\((\frac{n+1}{2})\) terms	\((\frac{n}{2})\) terms
(x+a)^n - (x-a)^n	\((\frac{n+1}{2})\) terms	\((\frac{n}{2})\) terms

Coefficient of	Binomial Expression	is
\((r+1)^{\text{th}}\)	\((1+x)^n\)	\({^nc_r}\)
\(x^r\)	\((1+x)^n\)	\({^nc_r}\)
\(x^r\)	\((1-x)^n\)	\((-1)^r\enspace{^nc_r}\)
\((r+1)^{\text{th}}\)	\((1-x)^n\)	\((-1)^r\enspace{^nc_r}\)

Operation	Notation
A or B	\(A \cup B\)
A and B	\(A \cap B\)
A but not B	\(A \cap \bar{B}\)
B but not A	\(\bar{A} \cap B\)
Neither A nor B	\(\bar{A} \cap \bar{B}\)
At least one of A,B,& C	\(A \cup B \cup C\)
Exactly one of A,B	\((A \cap \bar{B}) \cup (\bar{A} \cap B)\)
All Three of A,B,& C	\(A \cap B \cap C\)
Exactly Two of A,B,& C	\((A \cap B \cap \bar{C}) \cup (A \cap \bar{B} \cap C) \cup (\bar{A} \cap B \cap C)\)
Exactly One of A,B,& C	\((A \cap \bar{B} \cap \bar{C}) \cup (\bar{A} \cap B \cap \bar{C}) \cup (\bar{A} \cap \bar{B} \cap C)\)

Property 4
\(\sin^{-1}(\frac{1}{x}) = \cosec^{-1}(x)\)
\(\cos^{-1}(\frac{1}{x}) = \sec^{-1}(x)\)
\(\tan^{-1}(\frac{1}{x}) = \cot^{-1}(x),\quad x>0\)

Property 5
\(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\)
\(\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}\)
\(\sec^{-1}x + \cosec^{-1}x = \frac{\pi}{2}\)