The Cartesian coordinates of a point are given. a) (2,-2)

mattgondek4 2021-09-21 Answered
The Cartesian coordinates of a point are given.
a) (2,2)
b) (1,3)
Find the polar coordinates (r,θ) of the point, where r is greater than 0 and 0 is less than or equal to θ, which is less than 2π
Find the polar coordinates (r,θ) of the point, where r is less than 0 and 0 is less than or equal to θ, which is less than 2π
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Expert Answer

berggansS
Answered 2021-09-22 Author has 91 answers
Consider the Cartesian coordinates be (x,y).
The polar coordinates (r,θ) is defined as,
r2=x2+y2
x=rcosθ
y=rsinθ
This implies that, tanθ=yx
a) The Cartesian coordinates be (2,2).
Calculate r and θ as follows:
r2=22+(2)2
=8
r=±22
tanθ=yx
=22
=1
Thus, the angle θ is 3π4,7π4 as the angle 0θ2π
The polar coordinates of (2,-2) are,
(22,7π4), where r>0
(22,3π4), where r<0
b) The Cartesian coordinates be (1,3).
Calculate r and θ as follows:
r2=(1)2+(3)2
=4
r=±2
tanθ=yx
=31
=3
Thus, the angle θ is 2π3,5π3 as the angle 0θ2π
The polar coordinates of (1,3) are,
(2,2π3), where r>0
(2,5π3), where r<0
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