Polar Coordinates: Level 3 Challenges Practice Problems Online

If the center of a regular hexagon is at the origin and one of the vertices on the Argand plane is $1 + 2i$ , then what is its perimeter?

The Nautilus Spiral is given by the polar coordinates $r = e ^ \theta$ . Its shape resembles that of the Nautilus Shell, hence its name.

Which of the following statements about this curve is false?

The curve is self-similar. The half-line

\theta=\alpha

is cut into lengths which form a geometric progression. The distance along the curve from the origin to the point

\theta=0

is infinite. The curve from

\theta=-\infty

to

\theta=0

loops around the origin infinitely many times.

$\large \ln\left(\dfrac{\omega^\omega}{\omega^{\omega^2}}\right)$

If $\omega$ is a non-real complex cube root of unity, then what is the value of the expression above?

$\large \ln {( i) } = \ ?$

What do we obtain when we multiply the polar coordinates:

$( r_1 , \theta _1 ) \times ( r_2 , \theta _2 ) ?$

( r_1 \times r_2 , \theta_1 \times \theta _2 )

( r_1 + r_2 , \theta_1 \times \theta _2 )

( r_1 + r_2 , \theta_1 + \theta _2 )

( r_1 \times r_2 , \theta_1 + \theta _2 )

Polar Coordinates: Level 3 Challenges