Question

What is the polar form of `( -6,36 )`?

Answer 1

`" "`
**Cartesian Coordinates to Polar Form: ** `color(blue)((-6, 36) = (36, 99.4^@)`

Explanation:

`" "`
Given the Cartesian Form: `(-6, 36)`

Find the Polar Form:`color(blue)((r,theta)`

`color(green)("Step 1:"`

Let us examine some of the relevant formula in context:

`color(green)("Step 2:"`

Plot the coordinate point `color(blue)((-6, 36)` on a Cartesian coordinate plane:

Indicate the known values, as appropriate:

`bar(OA)=6" Units"`

`bar(AB)=36" Units"`

Let `bar(OB)=r" Units"`

`/_OAB=90^@`

Let `/_AOB=alpha^@`

`color(green)("Step 3:"`

Use the formula: `color(red)(x^2 + y^2=r^2` to find `color(blue)(r`

Consider the following triangle with known values:

`r^2=6^2+36^2`

`rArr 36+1296`

`rArr 1332`

`r^2=1332`

Hence, `color(brown)(r=sqrt(1332)~~36.4966`

To find the value of `color(red)(theta)`:

`tan(theta)=36/6=6`

`theta= tan^-1(6)`

`theta ~~ 80.53767779^@`

`color(blue)("Important:"`

Since the angle `color(red)(theta` lines in Quadrant-II, we must subtract this angle from `color(red)180^@` to get the required angle `color(blue)(beta`.

`color(green)("Step 4:"`

`color(blue)(beta ~~ 180^@ - 80.53767779^@`

`rArr beta ~~ 99.46232221^@`

Hence, the required Polar Form:

`color(blue)((r, theta) = (36, 99.4^@)`

Hope it helps.