9.) Transform each polar equation to an equation in rectangular coordinates and identify its shape:\
r = (4 / (2cosθ - 3sinθ));
10.) compute the modulus and argument of each complex number.
a.) -5
b. )-5 + 5i
9.)
\(r\ =\ \frac{4}{2\cos\theta-3\sin\theta}\\~\\ r(2\cos\theta-3\sin\theta)\ =\ 4\\~\\ 2r\cos\theta-3r\sin\theta\ =\ 4\\~\\ 2x-3y\ =\ 4\qquad\ \qquad\ \qquad\text{because}\qquad x=r\cos\theta\qquad\text{and}\qquad y=r\sin\theta\\~\\ 2x\ =\ 4+3y\\~\\ 2x-4\ =\ 3y\\~\\ \frac23x-\frac43\ =\ y\\~\\ y\ =\ \frac23x-\frac43\)
This is the equation of a line with a slope of \(\frac23\) and a y-intercept of \(-\frac43\) .
Check: https://www.desmos.com/calculator/7dq2bqym7k
(You can show or hide the second equation by clicking the gray circle to the left of it. )
9.)
\(r\ =\ \frac{4}{2\cos\theta-3\sin\theta}\\~\\ r(2\cos\theta-3\sin\theta)\ =\ 4\\~\\ 2r\cos\theta-3r\sin\theta\ =\ 4\\~\\ 2x-3y\ =\ 4\qquad\ \qquad\ \qquad\text{because}\qquad x=r\cos\theta\qquad\text{and}\qquad y=r\sin\theta\\~\\ 2x\ =\ 4+3y\\~\\ 2x-4\ =\ 3y\\~\\ \frac23x-\frac43\ =\ y\\~\\ y\ =\ \frac23x-\frac43\)
This is the equation of a line with a slope of \(\frac23\) and a y-intercept of \(-\frac43\) .
Check: https://www.desmos.com/calculator/7dq2bqym7k
(You can show or hide the second equation by clicking the gray circle to the left of it. )
10.) compute the modulus and argument of each complex number.
a.) -5
We have the form -5 + 0i
The modulus is √ [ (-50^2 + 0^2 ] = √25 = 5
The argument is θ so tan θ = 0 / -5 = pi
b. ) -5 + 5i
Modulus = √[ (-5)^2 + (5)^2 ] = √ [ 50] = 5√2
The argument is θ so tan θ = 5/-5 = - 1 = 3pi/4