Chapter 8.1, Problem 2E

(a)

To determine

To find: The value of $x$ and $y$ where $\left(x,y\right)$ are rectangular coordinates.

Answer to Problem 2E

The value of $x$ is $r\cos \theta$ and $y$ is $r\sin \theta$.

Explanation of Solution

Given:

The point $P$ is given with polar coordinates $\left(r,\theta \right)$.

Sketch a figure to show relationship between rectangular coordinates $\left(x,y\right)$ and polar coordinates $\left(r,\theta \right)$ as below.

Figure (1)

Figure (1) shows the relationship between rectangular coordinates $\left(x,y\right)$ and polar coordinates $\left(r,\theta \right)$.

Formula to find $x$.

$\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$

Substitute $x$ for adjacent and $r$ for hypotenuse in above formula.

$\begin{array}{c}\cos \theta =\frac{x}{r}\\ x=r\cos \theta \end{array}$

Formula to find $y$.

$\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$

Substitute $y$ for opposite and $r$ for hypotenuse in above formula.

$\begin{array}{c}\sin \theta =\frac{y}{r}\\ y=r\sin \theta \end{array}$

Thus the value of $x$ is $r\cos \theta$ and $y$ is $r\sin \theta$.

(b)

To determine

To find: The value of $r^2$ and $\tan \theta$ where $\left(r,\theta \right)$ are the polar coordinates.

Answer to Problem 2E

The value of $r^2$ is $x^2+y^2$ and $\tan \theta$ is $\frac{y}{x}$.

Explanation of Solution

Given:

The point $P$ is given with rectangular coordinates $\left(x,y\right)$.

Use Pythagorean Formula to find $r^2$ in figure (1)

${\left(\text{hypotenuse}\right)}^2={\left(\text{opposite}\right)}^2+{\left(\text{adjacent}\right)}^2$

Substitute $r$ for hypotenuse, $y$ for opposite and $x$ for adjacent in the above formula.

${\left(r\right)}^2={\left(y\right)}^2+{\left(x\right)}^2$

Formula to find $\tan \theta$ from figure (1)

$\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$

Substitute $y$ for opposite and $x$ for adjacent in the above formula.

$\tan \theta =\frac{y}{x}\left(x\ne 0\right)$

Thus the value of $r^2$ is $x^2+y^2$ and $\tan \theta$ is $\frac{y}{x}$.