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°
