Q&A #22: Parametric Equations & Polar Coordinates

Questions and Answers Sheet 22 Parametric Equations, Polar Coordinates, and Vector-V Question #1 What is 10< 30+10< 30? Answer in polar form. Note that the angle is measured in degrees here and need to be in Phasors. Answer: as they are on the same facing (parallel, but not anti-parallel) lines So 10<30+10<30 =20<30 Question #2 How do you find the polar coordinates of (−4, 0) Answer: The rectangular coordinates (𝑥, 𝑦) = (−4, 0) is equivalent to the polar coordinates (𝑟, 𝜃) = (4, 𝜋) Let us look at some details. Since r is the distance of (−4, 0) from the origin, 𝑟 = √(−4)2 + 02 = 4 Since the segment from the origin to (−4, 0) makes 180∘ = 𝜋 rad with the positive x-axis, 𝜃 = 𝜋. Question #3 How do you find the polar coordinates of the point (-2, 0)? Answer: The polar coordinates of (𝑥, 𝑦) = (−2, 0) 𝑖𝑠 (𝑟, 𝜃) = (2, 𝑝𝑖) Since the distance of the point from the origin is √(−2)2 + (0)2 = 2 so r=2 Since the line segment (vector) from the origin to (-2,0) and the positive x-axis makes 180∘ , which is 𝜋 radians; therefore, 𝜃 = 𝜋 Question #4 Find the polar coordinates of the point with rectangular coordinates (0, 1)? Answer: The polar coordinates of (x, y)=(0, 1) is 𝜋 (𝑟, 𝜃) = (1, ) 2 Since r is the distance of the point from the origin 𝑟 = √02 + 12 Since 𝜃 is the angle between the positive x-axis and the segment connecting the origin and the point, 𝜋 𝜃= 2 (Note: (0, 1) is on the positive y-axis, which makes 90∘ from the positive x-axis.) Question #5 𝜋 How do you find the rectangular coordinates of the point with polar coordinates (−1, ) ? 3 Answer: When trying to convert polar coords to rectangular coords or vice versa you need to be aware of a few equations. 𝑥 cos(𝜃) = → 𝑥 = 𝑟 cos(𝜃) sin(𝜃) = tan(𝜃) = 𝑟 𝑦 𝑟 𝑦 𝑥 → 𝑦 = 𝑟 sin(𝜃) 𝑦 → 𝜃 = tan−1 ( ) 𝑥 𝑥 2 + 𝑦 2 = 𝑟 2 → 𝑟 = √𝑥 2 + 𝑦 2 Polar Coordinates 𝜋 (𝑟, 𝜃) → (−1, ) 3 Calculate the x-value 𝑥 = 𝑟 cos(𝜃) 𝜋 𝑥 = (−1) × cos ( ) 3 1 𝑥 = (−1) ( ) 2 𝑥=− 1 2 Calculate the y-value 𝑦 = 𝑟 sin(𝜃) √3 𝑦 = (−1) ( ) 2 𝑦=− √3 2 Rectangular Coordinates 1 √3 2 2 (𝑥, 𝑦) → (− , − ) Question #6 𝜋 How do you find the rectangular coordinates of the point with polar coordinates (5, ) ? 2 Answer: To convert from polar coordinates to rectangular coordinates, we must remember these: 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃. It's useful to remember these when we need to do these types of conversions. Knowing the r and 𝜃 from our question, we can find the rectangular coordinates: r=5 𝜋 𝜃= 2 𝜋 𝑥 = (5) cos ( ) = (5)(0) = 0 2 𝜋 𝑦 = (5) sin ( ) = (5)(1) = 5 2 Therefore, our rectangular coordinates are: (0, 5) Question #7 How do you find the rectangular coordinates of the point with polar coordinates (3, 𝜋)? Answer: Since (𝑥, 𝑦) = (𝑟 cos 𝜃 , 𝑟 sin 𝜃) (𝑥, 𝑦) = (3 cos(𝜋) , 3 sin(𝜋)) = (−3,0) Question #8 1 4 What is ( , ) in polar coordinates? 2 5 Answer: Step 1 Use the formulas 𝑟2 = 𝑥 2 + 𝑦2, and 𝑦 tan 𝜃 = 𝑥 to change from rectangular to polar. Step 2 1 16 4 25 𝑟 = √𝑥 2 + 𝑦 2 = √ + = 0.94 4 5 1 2 𝑦 𝜃 = tan−1 ( ) = tan−1 ( ) = 1.01 𝑥 Question #9 𝜋 𝜋 4 3 How do you find the rectangular coordinates of the point whose spherical coordinates are (4, 3 , Answer: The formulas are: 𝑥 = 𝑟 sin 𝜃 cos 𝜙 𝑦 = 𝑟 sin 𝜃 sin 𝜙 𝑧 = 𝑟 cos 𝜃. So: 3 𝜋 √2 4 3 3 𝜋 2 √2 4 3 3 2 𝑥 = 4 sin ( 𝜋) cos ( ) = 4 × 𝑦 = 4 sin ( 𝜋) sin ( ) = 4 × 𝑧 = 4 cos ( 𝜋) = 4 × (− 4 √2 2 1 × = √2 × 2 √3 2 = √6 ) = −2√2 Question #10 How do you change (2√3, 2, − 1) from rectangular to cylindrical coordinates? Answer: First, you have to calculate the radius 𝑟 = √𝑥 2 + 𝑦 2 = √12 + 4 = 4 Second, you know that 𝑥 = 𝑟 cos(𝜃) and 𝑦 = 𝑟 sin(𝜃), therefore, cos(𝜃) = 𝜋 So, 𝜃 = (𝑚𝑜𝑑𝑢𝑙𝑜 2𝜋) 6 Conclusion, the cylindrical coordinates are (4, 𝜋 6 , − 1) √3 2 and 𝑦 = 1 2 )?