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Basic Mathematics & Measurements

Angles and Trigonometry

Angle Measurement & Conversion

Definition: An angle ($\theta$, কোণ) is defined as the ratio of the arc length ($S$, বৃত্ত চাপ) to the radius ($R$, ব্যাসার্ধ), expressed as $\theta =\frac{S}{R}$.

• Units: The primary units are Degrees and Radians.
• Conversions:
  • $1^c=\frac{180^{\circ }}{{\pi }^c}$
  • $1^{\circ }=\frac{{\pi }^c}{180^{\circ }}$
  • $180^{\circ }={\pi }^c$
• Sub-units of Degrees: $1^{\circ }=60^{\prime }=3600^{\prime \prime }$

Trigonometric Ratios & Identities

Core Ratios and Values: Based on a right-angled triangle with Perpendicular ($P$), Base ($B$), and Hypotenuse ($H$):

			$0^{\circ }$	$30^{\circ }$	$45^{\circ }$	$60^{\circ }$	$90^{\circ }$
$\sin \theta$	$\frac{P}{H}$	$\frac{1}{\csc \theta }$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos \theta$	$\frac{B}{H}$	$\frac{1}{\sec \theta }$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\tan \theta$	$\frac{P}{B}$	$\frac{1}{\cot \theta }$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	$ND$
$\cot \theta$	$\frac{B}{P}$	$\frac{1}{\tan \theta }$	$ND$	$\sqrt{3}$	$1$	$\frac{1}{\sqrt{3}}$	$0$
$\sec \theta$	$\frac{H}{B}$	$\frac{1}{cos\theta }$	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	$ND$
$\csc \theta$	$\frac{H}{P}$	$\frac{1}{\sin \theta }$	$ND$	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	$1$

• Pythagorean Identities:
  • ${\sin }^2\theta +{\cos }^2\theta =1$
  • $1+{\tan }^2\theta ={\sec }^2\theta$
  • $1+{\cot }^2\theta ={\csc }^2\theta$
• Additional Formulas:
  • $\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$
  • $\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$
  • $\tan (A\pm B)=\frac{\tan A\pm \tan B}{1\mp \tan A\tan B}$
  • $\sin 2\theta =2\sin \theta \cos \theta$
  • $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta =2{\cos }^2-1=1-2{\sin }^2\theta$
• Small Angle Approximation ($\theta <10^{\circ }$)
  When the Angle $\theta$ Is Very Small (Typically less than $10^{\circ }$), the Trigonometric Functions Simplify Drastically.
  • $\sin \theta \approx \theta$
  • $\tan \theta \approx \theta$
  • $\cos \theta \approx 1$
  • ! CRITICAL: For $\sin$ and $\tan$, $\theta$ Must Be Converted to Radians for This to Work!

Angles Greater than 90° (ASTC Rule)

• The quadrant system determines the sign of the trigonometric ratio:

  • Quadrant I: All positive.
  • Quadrant II: Sin and Cosec positive.
  • Quadrant III: Tan and Cot positive.
  • Quadrant IV: Cos and Sec positive.

• When calculating using $\boxed{90^{\circ }\times n\pm \theta }$:

  • $n$ is the Quadrant number.
  • If $n$ is even, the function remains the same (e.g., $\sin \leftrightarrow \sin$).
  • If $n$ is odd, the function changes (e.g., $\sin \leftrightarrow \cos$, $\sec \leftrightarrow \csc$, $\tan \leftrightarrow \cot$).

Negative Angles

Here $\cos$ and $\sec$ are positive with negative angles, while $\sin ,\csc ,\tan ,\cot$ remain negative.

• $\sin (-\theta )=-\sin (\theta )$
• $\cos (-\mathbf{\theta })=\cos (\mathbf{\theta })$
• $\tan (-\theta )=-\tan (\theta )$
• $\csc (-\theta )=-\csc (\theta )$
• $\sec (-\mathbf{\theta })=\sec (\mathbf{\theta })$
• $\cot (-\theta )=-\cot (\theta )$

The 37° / 53° Special Triangle

In a 3-4-5 right-angled triangle, the angle opposite to the side of length 3 is $37^{\circ }$, and the angle opposite to the side of length 4 is $53^{\circ }$. The hypotenuse is always 5. The angles are interchangeable

• For 37° Angle:
  • $\sin 37^{\circ }=\frac{P}{H}=\frac{3}{5}$
  • $\cos 37^{\circ }=\frac{B}{H}=\frac{4}{5}$
  • $\tan 37^{\circ }=\frac{P}{B}=\frac{3}{4}$
• For 53° Angle:
  • $\sin 53^{\circ }=\frac{P}{H}=\frac{4}{5}$
  • $\cos 53^{\circ }=\frac{B}{H}=\frac{3}{5}$
  • $\tan 53^{\circ }=\frac{P}{B}=\frac{4}{3}$

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Geometric Progression (GP)

A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the Common Ratio ($r$).

• Divergent Series: The terms keep growing (e.g., $2,4,8,16,32\dots$) and the sum tends to infinity.
• Convergent Series: The terms get progressively smaller (e.g., $1,\frac{1}{2},\frac{1}{4},\frac{1}{8}\dots$).
  • Common Ratio ($r$): $r=\frac{T_2}{T_1}=\frac{T_3}{T_2}$.
  • Sum of Infinite Convergent Series:
    $\boxed{S_{\infty }=\frac{a}{1-r}}$ (where $a$ is the first term and $|r|<1$).

Example Calculation

Q: Find the sum of the series: $h,e^{2}h,e^{4}h,e^{6}h\dots$ (where $e^2<1$)
Solution:

• $a=h$
• $r=e^2$
• $S=\frac{a}{1-r}=\frac{\mathbf{h}}{1-{\mathbf{e}}^2}$

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Binomial Approximation

When dealing with expansions of powers, we can use a simplified approximation if the second term is extremely small compared to $1$.

• Formula: $\boxed{(1\pm x{)}^n\approx (1\pm nx)}$ (Valid strictly when $x\ll 1$).

Practical Examples

• $(1.02{)}^{10}=(1+0.02{)}^{10}\approx 1+(10\times 0.02)=1+0.2=1.2$.
• $\sqrt{0.96}=(1-0.04{)}^{1/2}\approx 1-(\frac{1}{2}\times 0.04)=1-0.02=0.98$.

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Coordinate Geometry & Graphs

Slope of a Graph (গ্রাফের নতি)

The slope ($m$) measures the steepness and direction of a line.

• Formula: $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}=\tan \theta$.
• Angle Relation: If ${\theta }_3>{\theta }_2>{\theta }_1$, then ${\text{Slope}}_3>{\text{Slope}}_2>{\text{Slope}}_1$
• Types of Slope:
  

Key 2D Curves

Curve Type	General Equation	Additional Info
Straight Line	$y=mx+c$	$m$ = slope, $c$ = y-intercept.
Ellipse	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	Area = $\pi ab$ (where $a,b$ are major/minor semi-axes).
Circle	$x^2+y^2=R^2$	Area = $\pi R^2$ (Special case of ellipse where $a=b=R$).

Trigonometric Graphs

• Sine Curve ($y=\sin x$): Starts at origin $(0,0)$, reaches peak $+1$ at $\pi /2$, crosses zero at $\pi$, trough $-1$ at $3\pi /2$, and completes cycle at $2\pi$.
• Cosine Curve ($y=\cos x$): Starts at peak $+1$ at $0$, crosses zero at $\pi /2$, trough $-1$ at $\pi$, crosses zero at $3\pi /2$, and completes cycle at $2\pi$.

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Calculus: Differentiation (অবকলন)

Differentiation represents the rate of change of one physical quantity with respect to another.

• Velocity: $v=\frac{dx}{dt}$ (Rate of change of displacement).
• Acceleration: $a=\frac{dv}{dt}$ (Rate of change of velocity).

Standard Derivatives

• $\frac{d(c)}{dx}=0$ (where $c$ is a constant).
• $\frac{d(\sin x)}{dx}=\cos x$.
• $\frac{d(\cos x)}{dx}=-\sin x$.
• $\frac{d(\sec x)}{dx}=\sec x\tan x$.
• $\frac{d(\csc x)}{dx}=-\csc x\cot x$.
• $\frac{d(\tan x)}{dx}={\sec }^{2}x$.
• $\frac{d(\cot x)}{dx}=-{\csc }^{2}x$.
• $\frac{d(e^x)}{dx}=e^x$.
• $\frac{d(\ln x)}{dx}=\frac{1}{x}$.
• $\frac{d({\log }_{a}x)}{dx}=\frac{1}{x\ln a}$

Rules of Differentiation

1. Power Rule: $\frac{d(x^n)}{dx}=n{x}^{n-1}$.
2. Addition/Subtraction: $\frac{d}{dx}(u\pm v)=\frac{du}{dx}\pm \frac{dv}{dx}$.
3. Product Rule (UV): $\frac{d}{dx}(u\cdot v)=u\frac{dv}{dx}+v\frac{du}{dx}$.
4. Quotient Rule (U/V): $\boxed{\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}}$.
5. Chain Rule: Differentiating from the outside in and multiplying the results.
   • Ex: $\boxed{y=(\sin x{)}^2⟹\frac{dy}{dx}=2(\sin x{)}^{2-1}\cdot \cos x=2\sin x\cos x=\sin (2x}$.

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Calculus: Integration (সমাকলন)

Integration is mathematically the reverse process of differentiation. It is used to find areas under curves.

Types

• Indefinite: Includes a constant $c$.
• Definite: Evaluated between specific limits, no constant $c$.
• & ${\int }_0^{\frac{\pi }{2}}(\sin x+\cos x)=[\sin x-\cos x{]}_0^{\frac{\pi }{2}}=(\sin \frac{\pi }{2}-\cos \frac{\pi }{2})-(\sin 0-\cos 0)=2$

Standard Integrals

• $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$.
• $\int 1dx=x+c$.
• $\int \sin xdx=-\cos x+c$.
• $\int \cos xdx=\sin x+c$.
• $\int \frac{1}{x}dx=\ln x+c$.
• $\int e^{x}dx=e^x+c$.

The Constant of Integration

Always remember to add the constant of integration ($c$) for indefinite integrals.
Ex: $\int (3)dx=\int 3x^{0}dx=3\cdot \frac{x^1}{1}+c=3x+\mathbf{c}$.

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