Q&A #23: Parametric Equations & Polar Coordinates

Questions and Answers Sheet 23 Parametric Equations, Polar Coordinates, and Vector-V Question #1 How do you evaluate log 216 6? Answer: 62 = 36 63 = 36 × 6 = 180 + 36 = 216 1 So 2163 = 6 and log 216 6 = 1 3 Question #2 What is the distance between the following polar coordinates?: (4, 𝜋), (5, 𝜋) Answer: The distance formula for polar coordinates is 𝑑 = √𝑟12 𝑟22 − 2𝑟1 𝑟2 cos(𝜃1 − 𝜃2 ) Where d is the distance between the two points, 𝑟1 , and 𝜃1 are the polar coordinates of one point and 𝑟2 and 𝜃2 are the polar coordinates of another point. Let (𝑟1 , 𝜃1 ) represent (4, 𝜋) and (𝑟2 , 𝜃2 ) represent (5, 𝜋) ⇒ 𝑑 = √42 + 52 − 2 × 4 × 5 cos(𝜋 − 𝜋) ⇒ 𝑑 = √16 + 25 − 40 cos(0) ⇒ 𝑑 = √41 − 40 × 1 = √41 − 40 = √1 = 1 ⇒𝑑=1 Hence the distance between the given points is 1. Question #3 What is the Cartesian form of (−1, 4𝜋 3 )? Answer: We are given the polar form, so there is a radius and an angle. We want to convert to x, y coordinates. So we can use Euler's formula (or at least the idea behind it) to convert between Cartesian and polar: 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃. From that, we just plug in the numbers, remembering our unit circle: 4𝜋 1 3 2 cos ( ) = − 4𝜋 √3 3 2 and sin ( ) = − therefore 1 √3 (𝑥, 𝑦) = ( , 2 2 ) You could also notice that a negative radius is the same as adding or subtracting 𝜋 to the angle, hence (−1, 4𝜋 3 𝜋 ) = (1, ) 3 which I think is a bit easier to think about. Question #4 𝜋 What is the Cartesian form of (6, − ) ? 3 Answer: Answer: (3, −3√3) Explanation: 𝜋 The Cartesian coordinates (x, y) of the point (6, − ) ≡ (𝑟, 𝜃) are given as follows 3 𝑥 = 𝑟 cos 𝜃. 𝜋 = 6 cos (− ) 3 𝜋 = 6 cos ( ) =6⋅ 𝜋 3 3 =3 𝑦 = 𝑟 sin 𝜃. 𝜋 = 6 sin (− ) 3 𝜋 = −6 sin ( ) 3 −6 ⋅ √3 2 = −3√3 Therefore, the Cartesian coordinates are (3, −3√3) Question #5 𝜋 What is the Cartesian form of (6, − ) ? 6 Answer: Polar coordinates (𝑟, 𝜃) are equivalent to Cartesian Coordinates (𝑟 cos 𝜃 , 𝑟 sin 𝜃) 𝜋 Hence polar coordinates (6, − ) are equivalent to Cartesian Coordinates 6 𝜋 𝜋 (6 × cos (− ) , 6 × sin (− )) 6 6 𝜋 𝜋 i.e. (6 × cos ( ) , 6 × − sin ( )) 5 or (6 × √3 2 6 1 , −6 × ) 2 or (3√3, −3) Question #6 What is the Cartesian form of (12, 5𝜋 3 )? Answer: Explanation: Remember that polar coordinates are of the form (𝑟, 𝜃). Also, remember that our x-values correspond with cosine and y-values with sine. Then, remember that our sine and cosine values come from the unit circle, where r=1. So, when changing our coordinates from polar to cartesian coordinates we are taking (𝑟, 𝜃) → (𝑟 cos(𝜃) , 𝑟 sin(𝜃)). Notice that 5𝜋 3 𝜋 𝜋 3 3 = 2𝜋 − . So, we can say that 𝜃 = − , which is in the fourth quadrant. This means that cosine is positive, and sine is negative. Then we can essentially say that (12, 5𝜋 3 Question #7 What is the Cartesian form of (100, 7𝜋 12 𝜋 𝜋 1 √3 3 3 2 2 ) → (12 cos ( ) , −12 sin ( )) = (12 ( ) , −12 ( )) = (6, −6√3) )? Answer: 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃 Explanation: First, note that 7𝜋 is in Quadrant II which means the x-coordinate will be negative and the y-coordinate 12 positive. Using the identity above... 7𝜋 𝑥 = 100 cos ( ) ≈ −25.88 12 7𝜋 𝑦 = 100 sin ( ) ≈ 96.59 12 Hence, Cartesian Form: (-25.88, 96.59) Question #8 What is the distance between the following polar coordinates?: (2, 5𝜋 12 ) , (1, 3𝜋 12 ) Answer: Answer: 𝐷 = √5 − 2√3 ≈ 1.2393 Explanation: Distance between Polar Co-ordinates: 𝐴(𝑟1 , 𝜃1 ) and 𝐵(𝑟2 , 𝜃2 ) is 𝐷 = √𝑟12 + 𝑟22 − 2𝑟1 𝑟2 cos(𝜃1 − 𝜃2 ) … → (𝐼) We have, 𝑃1 (2, 5𝜋 12 ) and 𝑃2 (1, So, 𝑟1 = 2, 𝑟2 = 1, 𝜃1 = ⇒ 𝜃1 − 𝜃2 = 5𝜋 12 − 3𝜋 12 = 5𝜋 12 2𝜋 12 3𝜋 12 ) and 𝜃2 = = 𝜋 ⇒ cos(𝜃1 − 𝜃2 ) = cos(30∘ ) Using: (I) we get 6 3𝜋 12 = 30∘ 𝐷 = √22 + 12 − 2(2)(1) cos 3 0∘ ⇒ 𝐷 = √4 + 1 − 4 ⋅ √3 2 ⇒ 𝐷 = √5 − 2√3 ⇒ 𝐷 ≈ 1.2393 Question #9 What is the distance between the following polar coordinates?: (3, 7𝜋 12 ) , (1, 17𝜋 8 ) Answer: The distance formula for rectangular coordinates is 𝐷 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 Plugging in 𝑥𝑛 = 𝑟𝑛 cos(𝜃𝑛 ) and 𝑦𝑛 = 𝑟𝑛 sin(𝜃𝑛 ), expanding, and simplifying gives 𝐷 = √𝑟12 + 𝑟22 − 2𝑟1 𝑟2 cos(𝜃1 − 𝜃2 ) Plugging in (𝑟1 , 𝜃1 ) = (3, 7𝜋 12 ) and (𝑟2 , 𝜃2 ) = (1, 𝐷 = √32 + 11 − 2(3)(1) cos ( 𝐷 = √10 − 6 cos ( 𝐷 ≈ 3.04 37𝜋 24 ) 17𝜋 8 − 7𝜋 12 ) 17𝜋 8 ) Question #10 𝜋 5𝜋 8 8 What is the distance between the following polar coordinates?: (12, ) , (−8, ) Answer: Distance 𝑑 ≈ 14.4222 Explanation: Distance between two points knowing the polar coordinates is given by the formula using cosine rule 𝑑 = √𝑟12 + 𝑟22 − 2𝑟1 𝑟2 cos(𝜃2 − 𝜃1 ) 𝜋 𝑐 5𝜋 𝑐 Given 𝑟1 = 12, 𝑟2 = −8, 𝜃1 = ( ) , 𝜃2 = ( ) 8 8 𝑑 = √122 ± 82 − (2 ⋅ 12 ⋅ −8 ⋅ cos ( 5𝜋 8 𝜋 − )) 8   d = 208 + 192cos( )  14.4222, as cos( ) = 0 2 2 Question #10 What is the distance between the following polar coordinates?: (5, Answer: Given: 𝑟1 = 5 𝜃1 = − 5𝜋 12 𝑟2 = 5 11𝜋 𝜃2 = 6 The distance formula for polar coordinates is 𝐷 = √𝑟12 + 𝑟22 − 2𝑟1 𝑟2 cos(𝜃1 − 𝜃2 ) 𝐷 = √52 + 52 − 2(5)(5) cos (− = √25 + 25 − 50 cos (− 𝐷 = √50 + 50 √2 2 9𝜋 4 ≈ 9.2388 ) 5𝜋 12 − 11𝜋 6 ) −5𝜋 12 ) , (5, 11𝜋 6 ).