Answer:
a) (-3, 3)
(i) Polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ)
= (3√2, 0.75π)
(ii) Polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ)
= (-3√2, 1.75π)
b) (4, 4√3)
(i) Polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ)
= (8, 0.13π)
(ii) Polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ)
= (-8, 1.13π)
Step-by-step explanation:
We know that polar coordinates are related to (x, y) coordinates through
x = r cos θ
y = r sin θ
And r = √[x² + y²]
a) For (-3, 3)
(i) x = -3, y = 3
r = √[x² + y²] = √[(-3)² + (3)²] = √18 = ±3√2
If r > 0, r = 3√2
x = r cos θ
-3 = 3√2 cos θ
cos θ = -3 ÷ 3√2 = -(1/√2)
y = r sin θ
3 = 3√2 sin θ
sin θ = 3 ÷ 3√2 = (1/√2)
Tan θ = (sin θ/cos θ) = -1
θ = 0.75π or 1.75π
Note that although, θ = 0.75π and 1.75π satisfy the tan θ equation, only the 0.75π satisfies the sin θ and cos θ equations.
So, (-3, 3) = (3√2, 0.75π)
(ii) When r < 0, r = -3√2
x = r cos θ
-3 = -3√2 cos θ
cos θ = -3 ÷ -3√2 = (1/√2)
y = r sin θ
3 = -3√2 sin θ
sin θ = 3 ÷ -3√2 = -(1/√2)
Tan θ = (sin θ/cos θ) = -1
θ = 0.75π or 1.75π
Note that although, θ = 0.75π and 1.75π satisfy the tan θ equation, only the 1.75π satisfies the sin θ and cos θ equations.
So, (-3, 3) = (-3√2, 1.75π)
b) For (4, 4√3)
(i) x = 4, y = 4√3
r = √[x² + y²] = √[(4)² + (4√3)²] = √64 = ±8
If r > 0, r = 8
x = r cos θ
4 = 8 cos θ
cos θ = 4 ÷ 8 = 0.50
y = r sin θ
4√3 = 8 sin θ
sin θ = 4√3 ÷ 8 = (√3)/2
Tan θ = (sin θ/cos θ) = (√3)/4
θ = 0.13π or 1.13π
Note that although, θ = 0.13π and 1.13π satisfy the tan θ equation, only the 0.13π satisfies the sin θ and cos θ equations.
So, (4, 4√3) = (8, 0.13π)
(ii) When r < 0, r = -8
x = r cos θ
4 = -8 cos θ
cos θ = 4 ÷ -8 = -0.50
y = r sin θ
4√3 = -8 sin θ
sin θ = 4√3 ÷ -8 = -(√3)/2
Tan θ = (sin θ/cos θ) = (√3)/4
θ = 0.13π or 1.13π
Note that although, θ = 0.13π and 1.13π satisfy the tan θ equation, only the 1.13π satisfies the sin θ and cos θ equations.
So, (4, 4√3) = (-8, 1.13π)
Hope this Helps!!!
