Question

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

Short answer

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

Step-by-step solution

Equating the Two Equations

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

Solving for theta

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

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 \(\theta = 4\).

Converting from Polar to Rectangular Coordinates

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