Answer:
The polar coordinates are as follow:
a. (6,2π)
b. (18, π/3)
c. (2√2 , 3π/4)
d. (2, 5π /6)
Step-by-step explanation:
To convert the rectangular coordinates into polar coordinates, we need to calculate r, θ .
To calculate r, we use Pythagorean theorem:
r = [tex]\sqrt{ x^{2} +y^{2} }[/tex] ---- (1)
To calculate the θ, first we will find out the θ ' using the inverse of cosine as it is easy to calculate.
So, θ ' = cos ⁻¹ (x/r)
If y ≥ 0 then θ = ∅
If y < 0 then θ = 2 π − ∅
For a. (6,0)
Sol:
Using the formula in equation (1). we get the value of r as:
r = [tex]\sqrt{6^{2} + 0^{2} }[/tex]
r = 6
And ∅ = cos ⁻¹ (x/r)
∅ = cos ⁻¹ (6/6)
∅ =cos ⁻¹ (1) = 2π
As If y ≥ 0 then θ = ∅
So ∅ = 2π
The polar coordinates are (6,2π)
For a. (9,9/[tex]\sqrt{3}[/tex])
Sol:
r = 9 + 3(3) = 18
and ∅ = cos ⁻¹ (x/r)
∅ = cos ⁻¹ (9/18)
∅ = cos ⁻¹ (1/2) = π/3
As If y ≥ 0 then θ = ∅
then θ = π/3
The polar coordinates are (18, π/3)
For (-2,2)
Sol:
r =√( (-2)²+(2)² )
r = 2 √2
and ∅ = cos ⁻¹ (x/r)
∅ = cos ⁻¹ (-2/ 2 √2)
∅ = 3π/4
As If y ≥ 0 then θ = ∅
then θ = 3π/4
The polar coordinates are (2√2 , 3π/4)
For (-√3, 1)
Sol:
r = √ ((-√3)² + 1²)
r = 2
and ∅ = cos ⁻¹ (x/r)
∅ = cos ⁻¹ ( -√3/2)
∅ = 5π /6
As If y ≥ 0 then θ = ∅
So θ = 5π /6
The polar coordinates are (2, 5π /6)
