Question

Which quadrant is `(-4, -11pi/6)` in?

Answer 1

Quadrant `III`

Explanation:

To find a point `(r, theta)` in polar coordinates, imagine an arrow which rotates `theta` radians counterclockwise from the pole (what would be the positive `x` axis in rectangular coordinates) and then extends `r` in that direction.

Rotating negative radians counterclockwise is the same as rotating the same number clockwise. Thus, as `3pi/2` is a three-quarter rotation,`2pi` is a full rotation, and `3pi/2 < (11pi)/6 < 2pi`
rotating `(11pi)/6` in the clockwise direction will result in pointing to quadrant `I`

However, similarly, extending a negative value is the same as extending a positive value in the opposite direction. Thus the resulting arrow will be in quadrant `III`

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Alternatively, you could note that to convert from polar coordinates to rectangular, we let `x = rcos(theta)` and `y = rsin(theta)`

Then here,
`x = (-4)cos(-(11pi)/6) = -2sqrt(3)`
`y = (-4)sin(-(11pi)/6) = -2`

and `(-2sqrt(3), -2)` is in quadrant `III`