Answer:
a)
(i) The coordinates of the point in polar form is (5√2 , 7π/4)
(ii) The coordinates of the point in polar form is (-5√2 , 3π/4)
b)
(i) The coordinates of the point in polar form is (6 , π/3)
(ii) The coordinates of the point in polar form is (-6 , 4π/3)
Step-by-step explanation:
* Lets study the meaning of polar form
- To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):
1. r = √( x2 + y2 )
2. θ = tan^-1 (y/x)
- In Cartesian coordinates there is exactly one set of coordinates for any
given point
- In polar coordinates there is literally an infinite number of coordinates
for a given point
- Example:
- The following four points are all coordinates for the same point.
# (5 , π/3) ⇒ 1st quadrant
# (5 , −5π/3) ⇒ 4th quadrant
# (−5 , 4π/3) ⇒ 3rd quadrant
# (−5 , −2π/3) ⇒ 2nd quadrant
- So we can find the points in polar form by using these rules:
[r , θ + 2πn] , [−r , θ + (2n + 1) π] , where n is any integer
(more than 1 turn)
* Lets solve the problem
(a)
∵ The point in the Cartesian plane is (-5 , 5)
∵ r = √x² + y²
∴ r = √[(5)² + (-5)²] = √[25 + 25] = √50 = ±5√2
∵ Ф = tan^-1 (y/x)
∴ Ф = tan^-1 (5/-5) = tan^-1 (-1)
- Tan is negative in the second and fourth quadrant
∵ 0 ≤ Ф < 2π
∴ Ф = 2π - tan^-1(1) ⇒ in fourth quadrant r > 0
∴ Ф = 2π - π/4 = 7π/4
OR
∴ Ф = π - tan^-1(1) ⇒ in second quadrant r < 0
∴ Ф = π - π/4 = 3π/4
(i) ∵ r > 0
∴ r = 5√2
∴ Ф = 7π/4 ⇒ 4th quadrant
∴ The coordinates of the point in polar form is (5√2 , 7π/4)
(ii) r < 0
∴ r = -5√2
∵ Ф = 3π/4 ⇒ 2nd quadrant
∴ The coordinates of the point in polar form is (-5√2 , 3π/4)
(b)
∵ The point in the Cartesian plane is (3 , 3√3)
∵ r = √x² + y²
∴ r = √[(3)² + (3√3)²] = √[9 + 27] = √36 = ±6
∵ Ф = tan^-1 (y/x)
∴ Ф = tan^-1 (3√3/3) = tan^-1 (√3)
- Tan is positive in the first and third quadrant
∵ 0 ≤ Ф < 2π
∴ Ф = tan^-1 (√3) ⇒ in first quadrant r > 0
∴ Ф = π/3
OR
∴ Ф = π + tan^-1 (√3) ⇒ in third quadrant r < 0
∴ Ф = π + π/3 = 4π/3
(i) ∵ r > 0
∴ r = 6
∴ Ф = π/3 ⇒ 1st quadrant
∴ The coordinates of the point in polar form is (6 , π/3)
(ii) r < 0
∴ r = -6
∵ Ф = 4π/3 ⇒ 3rd quadrant
∴ The coordinates of the point in polar form is (-6 , 4π/3)
, which follows the formula: