The polar coordinates of a certain point are (r = 2.75 cm, θ = 219°). (a) Find its Cartesian coordinates x and y. x = cm y = cm (b) Find the polar coordinates of the points with Cartesian coordinates (−x, y). r = cm θ = ° (c) Find the polar coordinates of the points with Cartesian coordinates (−2x, −2y). r = cm θ = ° (d) Find the polar coordinates of the points with Cartesian coordinates (3x, −3y). r = cm θ = °

Answer:(a) x = 2.14 cm and y = 1.73 cm ⇒ the point is (-2.14 , -1.73)(b) r = 2.75 cm, Ф = 321.05°(c) r = 5.50 cm, Ф = 38.95(d) r = 8.26 cm, Ф = 141.05°Step-by-step explanation:* Lets explain the relation between the polar coordinates and the   Cartesian coordinates- The polar coordinates are (r , Ф°) - The Cartesian coordinates are (x , y)- [tex]r=\sqrt{x^{2}+y^{2}}[/tex] and Ф = [tex]tan^{-1}\frac{y}{x}[/tex]- x = r cosФ- y = r sinФ* Lets solve the problem- The polar form of the point is (r = 2.75 cm, θ = 219°)(a) ∵ r = 2.75 cm and Ф = 219°∵ x = r cosФ and y = r sinФ∴ x = 2.75 (cos 219) = -2.14∴ y = 2.75 (sin 219) = -1.73- The point (x , y) lies on the 3rd quadrant∴ x = 2.14 cm and y = 1.73 cm- We neglect (-) in the answer only because no negative values for cm(b) - We want to find the polar coordinates of (-x , y)- That means we want to find the polar coordinates of (2.14 , -1.73)∵ This points lies on the 4th quadrant∴ Ф will lies between 270° and 360°∵ [tex]r=\sqrt{x^{2}+y^{2}}[/tex]∴ [tex]r=\sqrt{(2.14)^{2}+(-1.73)^{2}}=2.75[/tex]∴ r = 2.75∵ α = [tex]tan^{-1}\frac{1.73}{2.14}=38.95[/tex], where α is acute angle∵ Ф lies in the 4th quadrant, then Ф = 360° - α∴ Ф = 360 - 38.95 = 321.05°∴ r = 2.75 cm, Ф = 321.05°(c) - We want to find the polar coordinates of (-2x , -2y)- That means we want to find the polar coordinates of (4.28 , 3.46)∵ This points lies on the 1nd quadrant∴ Ф will lies between 0° and 90°∵ [tex]r=\sqrt{x^{2}+y^{2}}[/tex]∴ [tex]r=\sqrt{(4.28)^{2}+(3.46)^{2}}=5.50[/tex]∴ r = 5.50∵ α = [tex]tan^{-1}\frac{3.46}{4.28}=38.95[/tex], where α is acute angle∵ Ф lies in the 1st quadrant, then Ф = α∴ Ф = 38.95∴ r = 5.50 cm, Ф = 38.95(d) - We want to find the polar coordinates of (3x , -3y)- That means we want to find the polar coordinates of (-6.42 , 5.19)∵ This points lies on the 2nd quadrant∴ Ф will lies between 90° and 180°∵ [tex]r=\sqrt{x^{2}+y^{2}}[/tex]∴ [tex]r=\sqrt{(-6.42)^{2}+(5.19)^{2}}=8.26[/tex]∴ r = 8.26∵ α = [tex]tan^{-1}\frac{5.19}{6.42}=38.95[/tex], where α is acute angle∵ Ф lies in the 2nd quadrant, then Ф = 180° - α∴ Ф = 180 - 38.95 = 141.05°∴ r = 8.26 cm, Ф = 141.05°