Question

We can describe the location of a point in the plane using different _________systems. The point shown in the figure has rectangular coordinates (_,_) and polar coordinates (_,_)

Short answer

Coordinate,rectangular coordinates are and polar coordinates are

$(\sqrt{2},\frac{\pi }{4})\&(-\sqrt{2},\frac{5\pi }{4})$

Step-by-step solution

Step 1. Given information

A point p in the coordinate system

Step 2. Concept used

The point p as shown in the figure, is in the coordinate system. We can use different coordinate systems for such as rectangular and polar. For the polar coordination system, we use the formula of $r^2$and $\tan \theta$.

Step 3. Calculation

A point can be located in both coordinate systems(rectangular & polar).

As a result, several coordinate systems may be used to represent the position of a point in the plane.

From the given graph, we can conclude that the rectangular coordinate is.$(1,1)$

For polar coordinates, use the formula$r^2=x^2+y^2$, where$(x,y)$are the rectangular coordinates.

$\begin{array}{l}\Rightarrow r^2=1^2+1^2\\ \Rightarrow r^2=1+1\\ \Rightarrow r^2=2\\ \Rightarrow r=\pm \sqrt{2}\end{array}$

Now use formula as$\tan \theta =\frac{y}{x}$ ,

$\begin{array}{l}\Rightarrow \tan \theta =\frac{y}{x}\\ \Rightarrow \tan \theta =\frac{1}{1}\\ \Rightarrow \theta =\frac{\pi }{4}\\ or\\ \Rightarrow \theta =\frac{5\pi }{4}\end{array}$

So, the polar coordinates are$(\sqrt{2},\frac{\pi }{4})\&(-\sqrt{2},\frac{\pi }{4})$ .