$\DeclareMathOperator{\asin}{asin}$ $\DeclareMathOperator{\acos}{acos}$ $\DeclareMathOperator{\atan}{atan}$ $\DeclareMathOperator{\taninv}{taninv}$
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# Section 11.2Coordinate Conversions

## Converting from Cartesian Coordinates to Polar Coordinates

There will be occasions when you need to convert from rectangular coordinates to polar coordinates.

A point’s distance from the pole is easy. If the point P has rectangular coordinates (ab), denoted hereafter as P:(ab), then, by the Pythagorean Theorem, $r= \sqrt{a^2+b^2}$. Or, defined as a function, if a point has coordinates (xy), then the point’s distance from the pole is $r(x,y)= \sqrt{x^2+y^2}$.

Determining a point's angle from the reference direction (the positive x-axis) is a bit trickier than determining its distance from the origin.

As illustrated in Figure 11.2.1 (a), if P:(ab) is in the 1st or 4th quadrant, then the angle from the reference direction has a measure of $\atan(b/a)$.

Figure 11.2.1 (a). If a point with coordinates $(a,b)$ is in the first or fourth quadrant,
its direction from the positive x-axis is $\atan(b/a)$.

However, there are two problems in determining a point's direction from the positive x-axis. The first problem arises when the point is is in the coordinate plane's 2nd or 3rd quadrant. The second problem arises when the point is on the y-axis.

Figure 11.2.1 (b) illustrates the first problem. It shows two points, $Q:(-1.75,-1.25)$ and $P:(1.75,1.25)$. The quotient of point Q’s coordinates is the same as the quotient of P’s coordinates (approximately 0.7142857), so the atan function cannot distinguish between points Q and P in terms of their directions from the positive x-axis.

Figure 11.2.1 (b). If a point with coordinates $(a,b)$ is in the 3rd quadrant, the quotient of its coordinates will be the same as the point having coordinates $(-a,-b)$ in the 1st quadrant. The quotient of the points' coordinates is the same, and therefore $\atan(a,b)$ and $\atan(-a,-b)$ will have the same value.

Figure 11.2.1 (c) shows a similar problem with using atan to distinguish between the direction of point $R:(-1.75,1.25)$ in the second quadrant from the direction of point $S:(1.75,-1.25)$ in the fourth quadrant. The quotient of S's coordinates is the same as the quotient of R’s coordinates. The value of $\atan\left(\frac{1.25}{-1.75}\right)$ is the same as the value of $\atan\left(\frac{-1.25}{1.75}\right)$ because both quotients are equal to $-\frac{1.25}{1.75}$.

Figure 11.2.1 (c). If a point with coordinates $(a,b)$ is in the 2nd quadrant, the quotient of its coordinates will be the same as the point having coordinates $(-a,-b)$ in the 4th quadrant. The quotient of the points' coordinates will be the same, and therefore $\atan(a,b)$ and $\atan(-a,-b)$ will have the same value.

In short, atan cannot distinguish between directions of points $(a,b)$ and $(-a,-b)$, nor can atan distinguish between directions of points $(a,-b)$ and $(-a,b)$.

The second problem arises if a point is on the y-axis. The function $\atan(\theta)$ (arctangent $\theta$) is defined only for values of $\theta$ such that $-\dfrac{\pi}{2}\lt \theta \lt \dfrac{\pi}{2}$. It is not defined for $\theta= \dfrac{\pi}{2}$ (when a point is on the positive y-axis) or for $\theta= -\dfrac{\pi}{2}$ (when a point is on the negative y axis).

A solution to both problems is to define a new function that we will call taninv (tangent inverse). It will have two independent variables: the point’s x-coordinate and the point's y coordinate.

Any value $\taninv(x,y)$ will be a point’s direction from the positive x-axis, given as an angle measure in radians between 0 and $2\pi$.

Figure 11.2.2. The taninv function gives the direction from the positive x-axis of a point with coordinates (x, y).

Figure 11.2.2 gives the definition of taninv as it appears in GC (click here for the GC file). The key idea is to pay attention to whether a point’s x-coordinate is positive, negative, or zero.

• If $x > 0$ and $y > 0$, then atan($y/x$) gives the correct direction (see Figure 11.2.1a).
• If $x > 0$ and $y < 0$, then atan$(y/x)$ will give a result between $-\pi/2$ and 0, so add $2\pi$ to give a result between $3\pi/2$ and $2\pi$.
• If $x < 0$, then the direction given by atan($y/x$) is off by a rotation of $\pi$ (see Figure 11.2.1b), so add $\pi$ to the value of atan($y/x$).
• If $x=0$ and $y > 0$, the point is on the positive y-axis. The point's direction from 0 (the reference direction) is $\pi/2$.
• If $x=0$ and $y < 0$, the point is on the negative y-axis. The point's direction from 0 (the reference direction) is $3\pi/2$.
• You should notice that taninv(0,0) is undefined. This makes sense because the point (0,0) does not have a direction from (0,0).

## Converting from Polar Coordinates to Rectangular Coordinates

The key idea in converting from polar to rectangular coordinates is that, in the unit circle, any point on the circle has coordinates $(\cos\theta,\sin\theta)$ where $\theta$ is the measure of the angle made by the point, (0,0), and the positive x-axis (Figure 11.2.3).

Figure 11.2.3. Polar coordinates $(1,\theta)$ converted to rectangular coordinates $(\cos\theta,\sin\theta)$.

The rectangular coordinates of points not on the unit circle will simply be the rectangular coordinates of the corresponding point on the unit circle, scaled by the point's distance from the origin (Figure 11.2.4)

Figure 11.2.4. Polar coordinates $(r,\theta)$ converted to rectangular coordinates $(r\cos\theta,r\sin\theta)$.

## Exercise Set 11.2.1

1. Enter the definition of taninv (Figure 11.2.2) in GC.
2. To do this, type "\taninv\ ctrl-9 x , y =" to start the definition. Type ctrl-shift-A to make the multipart definition. Type ctrl-shift-A to add a new line to the definition.

Test your definition with coordinates having directions (in radians) that you know.

3. After completing Exercise 11.0.5, enter taninv(0,0). GC will display "not a number". This is GC’s way of saying that taninv(0,0) is undefined.
1. Why is it reasonable that taninv(0,0) should be undefined? What would it mean were taninv(0,0) = 1?
2. What is it about the definition of taninv that keeps GC from giving a value for taninv(0,0)?

4. The following are points' coordinates in the polar system. Convert them from polar coordinates to rectangular coordinates.
1. $(3,2)$
2. $(-4,-1)$
3. $(\sqrt 7,\sqrt 7)$
4. $(0,5)$

5. The following are points' coordinates in the rectangular system. Convert them from rectangular coordinates to polar coordinates.
1. $(3,2)$
2. $(-4,-1)$
3. $(\sqrt 7,\sqrt 7)$
4. $(0,5)$
6. The set of points $(x,2)$ in rectanglar coordinates is the graph of $y=2$. Define a function f so that GC's dispayed graph of $r=f(\theta)$ in polar coordinates is the same as GC's displayed graph of $y=2$ in rectangular coordinates.

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