Question

Convert the rectangular coordinates (–6, 6) to polar coordinates.

Answer 1

The polar coordinates are (6(
`\sqrt[]{2}`, 3pi/4).

Rectangular coordinates are in the form of (x, y) and Polar coordinates are expressed in the form of (r,
`\theta`).

Relation between polar coordinates and rectangular coordinates-

x = r cos(
`\theta`), y = r sin(
`\theta`) and x^2 +y^2 =r^2 ...(1)

So by using above formulas we can solve our question.

Here , x= -6 and y= 6

r^2 = (-6)^2 +(6)^2

=72

=>r = 6(
`\sqrt[]{2}`)

Put the values of x and y in the mentioned formula in eq(1)

-6 = 6(
`\sqrt[]{2}` )cos
`\theta`

6 = 6(
`\sqrt[]{2}` )sin(
`\theta`),

=>-1/(
`\sqrt[]{2}` = cos(
`\theta`) , 1/
`\sqrt[]{2}`= sin
`\theta`

Here cos is negative and sin is positive so it lies in 2nd quadrant

so here
`\theta` lies between
`(\pi)/(2) \leq\theta\leq\pi`

`\theta`= π-π/4

=3π/4

So,(r,
`\theta`) = ( 6√2,
`(3\pi)/(4)` )

Answer 2

Answer: A Polar is (6√2,
`(3\pi )/(4)`)

Explanation:

Draw a line to point (see image). You need to find the length of that line and then the angle. Polar(length, angle)

Using pythagorean theorem

length² = (6)² + (-6)²

length² = 36 +36

length =√72

length =
`√(36 *2)`

length = 6√2

To find angle:

The triangle size is 6-6-6√2 Let's proportionally shrink so we can use unit circle numbers to figure angle. Becomes: 1-1-√2

So when we do sin x = opp/adj

sin x =
`(1)/(√(2) )` >get rid of radical on bottom

sin x =
`(√(2) )/(2)` > when is sin =
`(√(2) )/(2)` This happens at
`(\pi )/(4)` but we are in the 2nd quadrant so the angle is
`(3\pi )/(4)`

Polar is (6√2,
`(3\pi )/(4)`)