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Chapter 9: Problem 13

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

Short Answer

Expert verified
The point of intersection of the graphs of the given equations in rectangular coordinates is approximately (1.21, 1.21).

Step by step solution

01

Set Equations Equal

Since the two curves intersect when their \( r \) values are the same at the same angle \( \theta \), we can equate the two given equations i.e., \( 1 + \cos \theta = 1 - \sin \theta \) .
02

Solve the Trigonometric Equation

Subtracting 1 from both sides gives \( cos \theta = - sin \theta \). Then, dividing both the sides by \( \sqrt{2} \), we have \( \cos \theta / \sqrt{2} = - \sin \theta / \sqrt{2} \). This implies \( \cos \theta = - \sin \theta \). But we know that \( \cos (\pi/4 - \theta) = \sin \theta \), so we can equate this to \( - \sin \theta \) to find that \( \theta = \pi/4 \) .
03

Calculate the Radius and Convert to Rectangular Coordinates

Using the known value of \( \theta = \pi/4 \), we substitute it back to either of original equation let's say \( r = 1 + \cos \theta \), we get \( r = 1 + \cos(\pi/4) = 1 + 1/\sqrt{2} \) which is approximately equal to 1.707. The point of intersection in rectangular coordinates is calculated by \( x = r\cdot\cos \theta \) and \( y = r\cdot\sin \theta \), which gives us the point (1.21, 1.21).

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