Question

let (p) be a point in the plane.
(a) if (p) has polar coordinates ((r,\theta)) then it has rectangular coordinates ((x,y)) where (x=) blank and (y=) blank.
(b) if (p) has rectangular coordinates ((x,y)) then it has polar coordinates ((r,\theta)) where (r^{2}=) blank and (\tan(\theta)=) blank.
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Explanation:

Step1: Recall polar - rectangular conversion

The relationship between polar coordinates $(r,\theta)$ and rectangular coordinates $(x,y)$ is based on trigonometric functions. For a point $P$ with polar coordinates $(r,\theta)$, we use the right - triangle relationships in the coordinate plane.
$x = r\cos\theta$

Step2: Find $y$

Similarly, $y = r\sin\theta$

Step3: Convert from rectangular to polar for $r^{2}$

Using the Pythagorean theorem in the right - triangle formed by the $x$ and $y$ coordinates, $r^{2}=x^{2}+y^{2}$

Step4: Find $\tan\theta$

By the definition of the tangent function in a right - triangle, $\tan\theta=\frac{y}{x}$ ($x
eq0$)

Answer:

(a) $x = r\cos\theta$, $y = r\sin\theta$
(b) $r^{2}=x^{2}+y^{2}$, $\tan\theta=\frac{y}{x}$ ($x
eq0$)