Problem 72 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=3 \sin \left(\theta+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The graph of the function \(r=3 \sin \left(\theta +\frac{\pi}{4}\right)\) in polar coordinates is a rose curve rotated by \( \frac{\pi}{4} \) in the counter-clockwise direction.

Step 1: Understand the Polar Equation

Here we have a polar equation \(r=3 \sin \left(\theta +\frac{\pi}{4}\right)\) The function gives us the radius \( r \) as a function of angle \( \theta \) - that is, the distance from the origin at any given angle.

Step 2: Choose Values for \( \theta \) and Calculate Corresponding \( r \)

Now we choose several values for \( \theta \). It's often a good idea to start with \( \theta = 0 \), and increase by increments of \( \frac{\pi}{4} \) or \( \frac{\pi}{2} \) depending on the periodicity of the function. For each chosen \( \theta \), plug it into the equation to compute the corresponding \( r \). Note that as the \( sin \) function repeats every \( 2\pi \), we only need to consider \( \theta \) in range \([0, 2\pi]\)

Step 3: Plot the Points in Polar Coordinates

Next, plot these (r, \( \theta \)) pairs in polar coordinates. Remember that \( r \) is the distance from the origin and \( \theta \) is the angle from the positive x-axis. Plot all the calculated points and join them to get the graph of the function.

Step 4: Interpret the Graph

Finally, interpret the graph. The graph will be a rose curve, a shape that's common when graphing sine or cosine functions in polar coordinates, and the shift of \( \frac{\pi}{4} \) in the argument of \( sin \) function is causing a rotation of the graph.