Question

Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points. $$r=1 \text { and } r=\sqrt{2} \cos 2 \theta$$

Short answer

Question: Find the intersection points of the polar curves \(r = 1\) and \(r = \sqrt{2} \cos 2 \theta\). Answer: The intersection points can be found by setting the equations equal to each other and solving for \(\theta\), as well as plotting both curves and visually identifying any remaining intersection points. The algebraic solution for \(\theta\) is given by \(\theta = \frac{1}{2} \arccos\left(\frac{1}{\sqrt{2}}\right) + \pi k\), where \(k\) is an integer. Additionally, verify if there are other intersection points graphically by plotting both curves together.

Step-by-step solution

Set the equations equal to each other

Since we want to find where the two curves intersect, we need to find the values of \(r\) and \(\theta\) that satisfy both equations: $$r = 1 \text { and } r = \sqrt{2} \cos 2 \theta.$$ So, we set them equal to each other: $$1 = \sqrt{2} \cos 2 \theta.$$

Solve for \(\theta\) using algebraic methods

To find the possible values of \(\theta\) that satisfy the equation, we will first isolate \(\cos 2 \theta\): $$\cos 2 \theta = \frac{1}{\sqrt{2}}.$$ Next, use the inverse cosine function to find the angle that satisfies this equation: $$2 \theta = \arccos\left(\frac{1}{\sqrt{2}}\right).$$ Now, we want to find all the possible values of \(\theta\) that satisfy this equation, which can be done by considering the periodicity of the cosine function: $$\theta = \frac{1}{2} \arccos\left(\frac{1}{\sqrt{2}}\right) + \pi k \qquad \text{where } k \in \mathbb{Z}.$$

Plot the curves

Now, let's plot the polar curves \(r = 1\) and \(r = \sqrt{2} \cos 2 \theta\) together. You can do this using any graphing software or calculator that is capable of plotting polar. Visually identify the intersection points where both curves meet. Depending on the software used, you may need to adjust the step-size and domains for \(r\) and \(\theta\) in order to get an accurate plot.

Identify remaining intersection points using the graphical method

After plotting the curves, examine their intersection points. The algebraic method will give some solutions, while the graphical method will help us identify if there are other intersection points that could not be found algebraically. Compare the angles found in Step 2 to the angles that correspond to the intersection points on the graph. In conclusion, by using both algebraic and graphical methods, we can find the intersection points of the polar curves \(r = 1\) and \(r = \sqrt{2} \cos 2 \theta\).