Question

Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $$ r=8 \sin \theta ; r=4 \csc \theta $$

Short answer

The polar coordinates of the points of intersection are \((4\sqrt{2}, \frac{\pi}{4})\), \((4\sqrt{2}, \frac{3\pi}{4})\), \((-4\sqrt{2}, \frac{5\pi}{4})\), \((-4\sqrt{2}, \frac{7\pi}{4})\).

Step-by-step solution

Convert csc to sin

The second equation uses the cosecant function, which can be converted to sine for easier comparison. Using the identity \(\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)}\), we convert the second equation as follows: \[r = 4 \text{ csc } \theta \rightarrow r = \frac{4}{\text{sin}(\theta)}.\]

Rewrite the equation

Simplify the converted equation. Since \( r = \frac{4}{\text{sin}(\theta)} \), multiply both sides by \(\text{sin}(\theta)\) to get: \[r \text{sin}(\theta) = 4.\]

Combine the equations

We now have two equations: \(r = 8 \text{sin } \theta \) and \(r \text{sin}(\theta) = 4.\) Substitute \(r\) from the first equation into the second: \[8 \text{sin}^2(\theta) = 4.\]

Solve for Sin(θ)

To solve for \(\text{sin}(\theta)\), divide both sides of the equation by 8: \[\text{sin}^2(\theta) = \frac{4}{8} = \frac{1}{2}.\] Then take the square root of both sides: \[\text{sin}(\theta) = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}.\]

Find θ

Determine the angles where \(\text{sin}(\theta) = \frac{\sqrt{2}}{2} \) and \(\text{sin}(\theta) = -\frac{\sqrt{2}}{2}.\) These angles are \[\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}.\]

Calculate r for each θ

Substitute these \(\theta\) values back into \(r = 8 \text{sin } \theta \) to find the corresponding \(r\) values. For \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\): \[r = 8 \times \frac{\sqrt{2}}{2} = 4\sqrt{2}.\] For \(\theta = \frac{5\pi}{4}\) and \(\theta = \frac{7\pi}{4}\): \[r = 8 \times \left(-\frac{\sqrt{2}}{2}\right) = -4\sqrt{2}.\]

Write the polar coordinates

The polar coordinates of the points of intersection are: \[\left(4\sqrt{2}, \frac{\pi}{4}\right), \left(4\sqrt{2}, \frac{3\pi}{4}\right), \left(-4\sqrt{2}, \frac{5\pi}{4}\right), \left(-4\sqrt{2}, \frac{7\pi}{4}\right).\]

Key concepts

Trigonometric Identities

Trigonometric identities are essential tools when tackling problems in polar coordinates. They help simplify and transform equations, making them easier to work with. For instance, in our exercise, we transformed the cosecant function using the identity \text{csc}(\theta) = \frac{1}{\text{sin}(\theta)}\, turning a complex equation into a simpler one. Knowing and applying these identities swiftly can save a lot of time and effort in solving trigonometry problems. So, keep common identities like \(\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1\) at your fingertips.
By converting trigonometric functions, you can compare and combine equations more efficiently, just like in our problem where we converted and compared polar equations to find intersection points.

Graphing Polar Coordinates

Graphing polar coordinates might feel different at first, but it's quite engaging once you get the hang of it. Polar coordinates are based on radial distance and angles, making them perfect for circular and spiral patterns. In our problem, we graphed the equations \(r = 8 \text{ sin } \theta\) and \(r = 4 \text{ csc } \theta\) on the same polar grid. Here's a step-by-step approach:

• Identify the type of curve each equation represents, such as a circle, cardioid, or spiral.
• Break the angle \(\theta\) into increments to find corresponding \(r\) values.
• Plot the points and connect them smoothly.
• Label the curves for clarity.

Practicing with different equations will help you become proficient. Remember, the angle increases counterclockwise from the positive x-axis, just like in standard coordinates.

Solving Trigonometric Equations

Solving trigonometric equations in polar form involves a detailed process of transformation and simplification. For example, we simplified \(r = 4 \text{ csc } \theta\) into \(r = \frac{4}{\text{sin}(\theta)}\). This transformation allowed us to easily juxtapose it with \(r = 8 \text{ sin } \theta\).

• First, express all terms using common trigonometric functions.
• Combine equations when required, as we did by setting \(r \text{ sin}(\theta) = 4\).
• Solve for the trigonometric function first, such as finding \(\text{sin}(\theta)\) values.
• Finally, solve for the variable \(\theta\).

With practice, these steps become second nature, making the solving process feel intuitive and manageable.

Intersection Points

Finding intersection points in polar coordinates involves both solving the equations and understanding their graphical representation. From the solution: We first simplified the equations and then solved for \(\text{sin}(\theta)\) to find suitable angles \(\theta\). Substituting these angles back yielded the radial distances \(r\).

• Identify angles where both equations meet.
• Calculate the corresponding \(r\) for these angles.
• The intersection points are your polar coordinates of those \(r\) and \(\theta\) pairs.

In our exercise, we found: \((4\sqrt{2}, \frac{\pi}{4})\), \((4\sqrt{2}, \frac{3\pi}{4})\), \((-4\sqrt{2}, \frac{5\pi}{4})\), \((-4\sqrt{2}, \frac{7\pi}{4})\). Understanding these steps deeply ensures accuracy in your answers and solidifies your grasp of polar coordinates.