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Chapter 9: Problem 17

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=\frac{\theta}{2} \\ r=2 \end{array} $$

Short Answer

Expert verified
The points of intersection of the graphs of the equations \(r = \frac{\theta}{2}\) and \(r = 2\) are (2cos4, 2sin4)

Step by step solution

01

Equating the Two Equations

To find the points of intersection, equate the two equations. This yields: \(\frac{\theta}{2} = 2\)
02

Solving for theta

Next, solve for \(\theta\) by multiplying both sides by 2. This gives: \(\theta = 4\)
03

Substituting theta Value in Both Equations

Substitute \(\theta = 4\) into both equations to get their intersecting points in terms of r. The results are, for \(r = \frac{\theta}{2}\) we have \(r = 2\) and for \(r = 2\) it remains \(r = 2\). These two results are the same, proving these two graphs intersect when \(r = 2\) and \(\theta = 4\).
04

Converting from Polar to Rectangular Coordinates

Finally, convert the polar coordinates (r, 胃) to rectangular coordinates (x, y). Recall that x = rcos胃 and y = rsin胃. Plugging in the values r = 2 and 胃 = 4 gives (x, y) as (2cos4, 2sin4)

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