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Explain why the point (r,θ)=8,-π3is not a polar representation of the point with rectangular coordinates (x,y)=(-4,43), even though these values of r,θ,x, and ysatisfy the formulas r2=x2+y2and tanθ=yx. Include a picture with your explanation.

Short Answer

Expert verified

The required answer is (-4,43)is in the second quadrant where as point localid="1652685710587" 8,-π3is in the fourth quadrant

Step by step solution

01

Given information

The rectangular coordinate (x,y)=(-4,43)

02

Simplification

Consider the rectangular coordinate (x,y)=(-4,43)

The objective is to find a polar representation.

In the coordinate (-4,43),x=-4and y=43.

To calculate θuse the formula θ=tan-1yx.

Then

θ=tan-143-4[sincex=-4,y=43]θ=tan-1(-3)θ=-π3sincetan-π3=-tanπ3=-3

To find the value of ruse the equation r=x2+y2.

Then,

r=(-4)2+(43)2[sincex=-4,y=43]r=16+48r=64r=±8

Then (r,θ)=8,-π3.

But the point 8,-π3is in the fourth quadrant. whereas (-4,43)is in the second quadrant.

The correct representation of the point (-4,43)is -8,5π3.

Since every rectangular point will have infinitely many representations in polar plane.

Thus, the coordinate (-4,43)is in the second quadrant 8,-π3is in the fourth quadrant. That is why the point 8,-π3is not a polar representation for the point (-4,43).

Therefore, the required answer is (-4,43)is in second quadrant where as point (8,-3)is in the fourth quadrant

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