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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

 May 28, 2019

Best Answer 

 #1
avatar+8756 
+3

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. )

 May 28, 2019
 #1
avatar+8756 
+3
Best Answer

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. )

hectictar May 28, 2019
 #2
avatar+103917 
+1

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 

 

 

cool cool cool

 May 29, 2019

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