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Chapter 9: Problem 16

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=1+\cos \theta \\ r=3 \cos \theta \end{array} $$

Short Answer

Expert verified
The points of intersection for the two equations are \((1.5, 60掳)\) and \((1.5, 300掳)\), or in radians, \((1.5, \pi/3)\) and \((1.5, 5\pi/3)\).

Step by step solution

01

Set the Equations Equal to Each Other

In order to find the points of intersection, you first set the two equations equal to each other. This gives us: \(1 + \cos \theta = 3 \cos \theta\).
02

Rearrange the Equation

Rearrange the equation to isolate the cosine function. The equation becomes: \(\cos \theta = 0.5\).
03

Solving for theta

To solve for 胃, use the arccos or inverse cosine function to get: \(\theta = \arccos(0.5)\). This yields \(\theta = 60掳\) and \(\theta = 300掳\) (or in radians, \(\theta = \pi/3\) and \(\theta = 5\pi/3\)).
04

Substituting Back for r

Substitute the calculated 胃 values back into either of the original equations to get the corresponding r values. By substituting the 胃 values in the second equation \(r = 3 \cos \theta\), for \(\theta = 60掳\), \(r = 3 \cos 60掳 = 1.5\) and for \(\theta = 300掳\), \(r = 3 \cos 300掳 = 1.5\).

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