Basic Math for Physics

Basic Mathematics for Physics

Tags: #physics #basicmath #calculus #trigonometry #boardprep

Angles and Trigonometry (কোণ এবং ত্রিকোণমিতি)

Understanding angles and their mathematical relationships is foundational for mechanics and vectors.

Angle Measurement & Conversion (কোণ এবং তার একক)

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 (ত্রিকোণামাত্রিক অনুপাত)

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 T rigonometric 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 $90^{\circ }\times n\pm \theta$:
  • $n$ 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.

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

The 37° / 53° Special Triangle (37° / 53° বিশেষ সমকোণী ত্রিভুজ)

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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Algebra & Series Expansions

Geometric Progression / GP (গুণোত্তর প্রগতি)

A sequence where each term is multiplied by a constant value called the Common Ratio (সাধারণ অনুপাত), denoted by $p$ or $r$.

Common Ratio: A common ratio is the constant multiplier between consecutive terms in a geometric sequence, found by dividing any term by the one immediately preceding it.

• Formula: $r=\frac{T_{(n+1)}}{T_n}$

Types of Series

Divergent Series (অপমারি)

A series diverges when its running totals do not settle on a single real number, instead growing without end or bouncing around.

• Ex: 2, 4, 8, 16…$\infty$.
• Ex: $1-1+1-1+\dots$
• Sum of an Infinite Divergent Series: A divergent series is an infinite sum that fails to settle on a single, finite number. This happens either because the partial sums grow larger and larger without a “ceiling” (tending toward infinity) or because they oscillate back and forth between different values. Because the total never converges to one specific point, mathematicians say the sum does not exist (DNE).

Convergent Series (অভিমারি)

A series converges when its running totals (called partial sums) get closer and closer to a fixed, real number.

• E.g. 1, 1/2, 1/4, 1/8…$\infty$

• Sum of an Infinite Convergent Series: When the common ratio is less than 1, the sum to infinity is $\frac{a}{1-r}$, ***where $a$ is the first term.

• Sum of an Infinite Divergent Series: A divergent series is an infinite sum that fails to settle on a single, finite number. This happens either because the partial sums grow larger and larger without a “ceiling” (tending toward infinity) or because they oscillate back and forth between different values. Because the total never converges to one specific point, mathematicians say the sum does not exist (DNE).

• Sum of an Infinite Convergent Series: When the common ratio is less than 1, the sum to infinity is $\frac{a}{1-r}$, where $a$ is the first term.

Binomial Expansion (দ্বিপদ উপপাদ্য)

Used to simplify complex powers when the value of $x$ is very small compared to 1.

• Formula: $(1\pm x{)}^n\approx (1\pm nx)$

• Example Application: $\sqrt{0.96}=(1-0.04{)}^{1/2}\approx 1-(\frac{1}{2}\times 0.04)=0.98$.

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3. Coordinate Geometry and Graphs

Slopes (গ্রাফের নতি)

Definition: The slope ($m$) of a graph represents its steepness and is calculated as $\frac{y_2-y_1}{x_2-x_1}$ aka $\tan \theta$ cause we’re dividing the perpendicular of a triangle by its base. $m\propto \theta$

• Types of Slopes (বিভিন্ন ধরনের নতি):

  • Positive (+ve): Angle $<90^{\circ }$, graph is increasing.
  • Negative (-ve): Angle $>90^{\circ }$, graph is decreasing.
  • Zero: Angle is $0^{\circ }$ ($\tan 0^{\circ }=0$), graph is constant/horizontal.
  • Constant: Angle remains unchanged at every point, forming a perfectly *straight line.
  • Infinite/Undefined: Angle is $90^{\circ }$ ($\tan 90^{\circ }=\infty$), graph is vertical.

Common 2D Shapes and Equations

• Straight Line (সরল রেখার গ্রাফ): $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept (where the line crosses the vertical axis)

• Ellipse (উপবৃত্ত): $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a$ is the semi-major axis and $b$ is the semi-minor axis. The variables $x$ and $y$ represent the coordinates of any point located exactly on the boundary or curve of the ellipse. But the equation is kinda a rule. So, if you put a value for $x$ it decides the corresponding $y$ value and the other way around.

  • Area: $\pi ab$

• Circle (বৃত্ত): $x^2+y^2=r^2$, circle is just a special type of ellipse where the two axes are equal. So, the $x,y$ rules apply.

Trigonometric Graphs (ত্রিকোণমিতির গ্রাফ)

• $y=\sin x$: Starts at 0, reaches +1 at $\frac{\pi }{2}$, crosses 0 at $\pi$, drops to -1 at $\frac{3\pi }{2}$, and returns to 0 at $2\pi$.

Graph: y = sin(x)

bounds: [-10, 5, 10, -5]
elements: [
  {type: functiongraph, def: ["f:Math.sin(x)"]}
]

• $y=\cos x$: Starts at +1 at 0, drops to 0 at $\frac{\pi }{2}$, reaches -1 at $\pi$, and returns to +1 at $2\pi$.

Graph: y = cos(x)

bounds: [-10, 5, 10, -5]
elements: [
  {type: functiongraph, def: ["f:Math.cos(x)"]}
]

4. Basic Calculus

Differentiation (অবকলন)

Represents the infinitesimally small change in $y$ with respect to $x$, denoted as $\frac{dy}{dx}$. It measures the Rate of Change. For example, velocity is the rate of change of displacement ($v=\frac{dx}{dt}$).

• Basic Rules:

  • Constant Rule: $\frac{d(a)}{dx}=0$
  • Addition/Subtraction Rule:$\frac{d(u\pm v)}{dx}=\frac{du}{dx}\pm \frac{dv}{dx}$
  • Power Rule: $\frac{d(x^n)}{dx}=n{x}^{n-1}$
    • ! Any number preciding $x$ will stay there like $\frac{d6x^2}{dx}=6\cdot 2\cdot x^{2-1}$
    • & It works for integration too
  • Product Rule: $\frac{d(u\cdot v)}{dx}=u\cdot \frac{dv}{dx}+v\cdot \frac{du}{dx}$
  • Division Rule: $\frac{d\left(\frac{u}{v}\right)}{dx}=\frac{v\cdot \frac{du}{dx}-u\cdot \frac{dv}{dx}}{v^2}$

• Chain Rule: Used for composite functions, differentiating, and multiplying from the outside in. Examples Below:

  • $\frac{d}{dx}\sin (x^2+5)=\cos (x^2+5)\cdot (2x)$
  • $\frac{d(\sin x{)}^2}{dx}=2\sin x^{2-1}\cdot \cos x$
  • $\frac{d}{dx}(4\sin 3x)=\frac{d}{dx}(4\sin 3x)\cdot \frac{d}{dx}3x$ (where u being the inner function)

• Standard Derivatives:

  • $\frac{d}{dx}(\sin x)=\cos x$
  • $\frac{d}{dx}(\cos x)=-\sin x$
  • $\frac{d}{dx}(\sec x)=\sec x\cdot \tan x$
  • $\frac{d}{dx}(\csc x)=-\csc x\cdot \cot x$
  • $\frac{d}{dx}(\tan x)={\sec }^{2}x$
  • $\frac{d}{dx}(\cot x)=-{\csc }^{2}x$
  • $\frac{d}{dx}(e^x)=e^x$
  • $\frac{d}{dx}(\ln x)=\frac{1}{x}$

Integration (সমাকলন)

The reverse process of differentiation. It is physically interpreted as finding the area under the curve.

• 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 (সমাকলনের নিয়মাবলী):

  • Power Rule: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$
  • $\int dx=x+c$
  • $\int \sin xdx=-\cos x+c$
  • $\int \cos xdx=\sin x+c$
  • $\int \frac{1}{x}dx=\ln x+c$
  • $\int e^{x}dx=e^x+c$
  • $\int x^{2{\ln }_{10}100}$