91Ó°ÊÓ

Binomial cubes: \((A+B)^{3}=A^{3}+3 A^{2} B+3 A B^{2}+B^{3}\) The cube of any binomial can be found using the formula shown, where \(A\) and \(B\) are the terms of the binomial. Use the formula to compute \((1-2 i)^{3}\) (note \(A=1\) and \(B=-2 i\) ).

$$ r=4 \sin 2 \theta $$

$$ r^{2}=9 \sin (2 \theta) $$

The midpoint formula in polar coordinates: $$ M=\left(\frac{r \cos \alpha+R \cos \beta}{2}, \frac{r \sin \alpha+R \sin \beta}{2}\right) $$ The midpoint of a line segment connecting the points \((r, \alpha)\) and \((R, \beta)\) in polar coordinates can be found using the formula shown. Find the midpoint of the line segment between \((r, \alpha)=\left(6,45^{\circ}\right)\) and \((R, \beta)=\left(8,30^{\circ}\right)\), then convert these points to rectangular coordinates and find the midpoint using the "standard" formula. Do the results match?