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Chapter 6: Problem 24

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(\sqrt{3},-1)$$

Short Answer

Expert verified
The polar coordinates of the point \((\sqrt{3}, -1)\) are \((2, 5蟺/6)\)

Step by step solution

01

Compute the polar distance

We need to find the distance of the point from the origin. For rectangular coordinates \((x, y)\), the polar distance r can be computed as \(r = \sqrt{x^2 + y^2}\). Plugging the given coordinates into this formula gives \(r = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = 2\). Thus, the polar distance of the point from the origin is 2.
02

Compute the polar angle

Now, need to find the polar angle. The polar angle 胃 is found by computing the arctangent of the ratio of the y and x coordinates, i.e., \(胃 = arctan(y/x)\). For our point, this gives \(胃 = arctan((-1)/\sqrt{3}) = -蟺/6\). Considering the signs of the coordinates, our point is in the fourth quadrant, so we add 蟺 to ensure the angle is within the bounds of \([-蟺, 蟺]\) obtaining that \(胃 = -蟺/6 + 蟺 = 5蟺/6\). Thus, the polar angle of the point is \(5蟺/6\).
03

Write the polar coordinates

The polar coordinates of a point are given by the pair \((r, 胃)\) representing the polar distance and the polar angle respectively. For the point, from step 1 and step 2 we have \(r = 2\) and \(胃 = 5蟺/6\). Therefore, the polar coordinates of the point are \((2, 5蟺/6)\).

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Most popular questions from this chapter

\(r=2(1+\cos \theta)\)

\(r=4(1+\sin \theta)\)

In the polar coordinate system, if a graph that has symmetry with respect to the pole were folded on the line \(\theta=3 \pi / 4\), the portion of the graph on one side of the fold would coincide with the portion of the graph on the other side of the fold.

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Hyperbola } & (2,0),(8,0) \\ \end{array} $$

\(r=3+6 \sin \theta\)
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