Thomas Calculus | Exercise 10.5 | Question 49-62 | Cartesian to Polar Equations | Allah Dad

Thomas Calculus | Exercise 10.5 | Question 49-62 | Cartesian to Polar Equations | Allah Dad Chapters: (00:00) Introduction (01:58) Question NO 49 (02:23) Question NO 50 (02:45) Question NO 51 (03:37) Question NO 52 (04:06) Question NO 53 (04:27) Question NO 54 (05:28) Question NO 55 (06:28) Question NO 56 (07:14) Question NO 57 (07:59) Question NO 58 (09:28) Question NO 59 (10:19) Question NO 60 (11:05) Question NO 61 (12:15) Question NO 62 📘 Thomas Calculus 11th Edition | Chapter 10: Conic Sections and Polar Coordinates 🎯 Exercise 10.5 | Questions 49 to 62 – Cartesian to Polar Equations 👨‍🏫 Instructor: Allah Dad In this video, we solve Exercise 10.5 Questions 49 to 62 from Thomas Calculus 11th Edition, focusing on the concept of polar coordinate pairs. we solve Questions 49 to 62 from Exercise 10.5, which involve converting points from polar form to Cartesian form using the formulas: x = rcos(theta) , y=rsin(theta) we solve Questions 49 to 62 from Exercise 10.5, which involve graphing polar equations and identifying the type of conic section represented by each polar equation. We explain: ✅ Step-by-step explanations with clear visuals and logic ✅ Step-by-step calculations with trigonometric values ✅ Square roots and angle simplifications ✅ Real-world interpretation of coordinate systems ✅ Helpful for exams and graphing problems in polar form ✅ How to Convert Carteian Equations to Polar Equations ✅ Use of polar equations like ✅ Meaning of eccentricity (e) and how it defines the conic shape ✅ How to sketch the graph of polar equations step-by-step ✅ How to find focus, directrix, and axis from polar form ✅ Tips for quickly recognizing and sketching polar graphs This video is perfect for students studying: 🧠 Conic Sections, Polar Coordinates, or Cartesian Conversions 📚 BS Math, Engineering, Class 12, and University Calculus, Engineering Students Anyone studying polar coordinate systems, conic sections, or parametric equations. This is essential content for students learning: Polar coordinates, Conic sections in polar form, Graph sketching techniques for calculus and analytical geometry 📘 Concept Explanation: (Polar Coordinate Pairs) What are Polar Coordinates? Polar coordinates represent a point using a radius (r) and an angle (θ) from the origin. Instead of (x, y), we use (r, θ). 🌀 Why Do Polar Coordinate Pairs Matter? In polar form: A single point can have many different polar coordinates (due to rotations and sign changes in r). For example, (3, 0), (−3, π), and (3, 2π) all represent the same point on the x-axis. This helps in rotational symmetry problems and in graphing conics and spirals easily. 🔄 Purpose: Helps graph curves like spirals, circles, cardioids easily. Simplifies many problems in physics, engineering, and math modeling. Used in converting between polar and Cartesian systems 🎓 Topics Covered / Keywords: Thomas Calculus Exercise 10.5, Convert Polar to Rectangular Coordinates, Polar Coordinates Explained, Cartesian Coordinates, r cos θ and r sin θ, Polar Graphs, Math Problem Solving, Engineering Calculus, Graphing Polar Equations, Conic Sections in Polar Form, Eccentricity in Conics, Polar Coordinate Graphs, Ellipse from Polar Equation, Hyperbola from Polar Form, Parabola in Polar Coordinates, Focus Directrix Form, Calculus Chapter 10 Solutions, Coordinate Geometry, Coordinate System Conversion, Polar Coordinates, Polar Coordinate Pairs, Graphing in Polar Form, Conic Sections, θ and r in Polar Equations, Equivalent Polar Points, Thomas Calculus 11th Edition, Chapter 10 Solutions, Polar to Cartesian, Engineering Math, Calculus Q49 to Q62 Solved 📖 This video is based on Thomas' Calculus (11th Edition) by George B. Thomas, Maurice D. Weir, and Joel Hass – published by Pearson Education. #thomascalculus #polarequations #polartocartesian #mathhelp #cartesianequations #AllahDad 🔔 Subscribe for full chapter solutions and conceptual lectures from Thomas Calculus!