11-1 Polars

This week in advanced math we learned chapter 11. We started by learning all of the different Polar graphs. There is Circles in Polar form where the formula is r= a cos(theta) or r= a sin(theta). Limacons where r=a +/- b cos(theta) or sin(theta) where a>0 and b>0. Cardiods where r=a +/- a sin(theta) or cos(theta). Rose Curves where r= a sin n(theta) or a cos n(theta). Lemniscates where r^2= a^2 sin 2(theta) or cos 2(theta). Then there's 11-2 with complex numbers. Which involves the formulas z= x + yi for rectangular and z= rcos(theta) + rsin(theta)i for Polar. The abbreviated version of this formula is z=rcis(theta). 11-3 involves De Moirre's Theorem with the formula (rcis)^n = r^n cisn(theta).

11-1 Polars

Formulas: x=rcos(theta); y=rsin(theta) to convert to rectangular

r=the square root of x^2 + y^2; tan(theta)=y/x to convert to polar

Ex. convert 2, 45 degrees to rectangular

x= 2cos45 degrees = 2(square root of 2)/2= square root of 2

y=2sin45degrees= 2(square root of 2)/2 = square root of 2

[square root of 2, square root of 2]