Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the rotated conic. $$r=\frac{3}{1-\cos (\theta-\pi / 4)}$$

Step-by-Step Solution

Answer

The graph is a conic section (ellipse, parabola, or hyperbola) rotated by an angle \(\pi /4\). The exact shape of the conic section would depend on the specific points you select, which might be guided by constraints given by the learning objective or the software or hardware in use.

Step 1: Understand Polar Coordinates

In polar coordinates, a point in a plane is determined by its distance from a reference point, \(r\), and the angle \(\theta\) from a reference direction. For this problem, the polar function is given as \(r =\frac{3}{1-\cos(\theta-\pi /4)}\).

Step 2: Convert Polar to Cartesian Coordinates

Each polar coordinate \((r, \theta)\) corresponds to the Cartesian coordinate \((r\cos\theta, r\sin\theta)\). Apply this conversion to a number of points determined by varying the value of \(\theta\).

Step 3: Plot Points on Graphing Utility

On the graphing utility, plot the Cartesian coordinate points found from step 2. Varying \(\theta\) should give a variety of points to plot.

Step 4: Connect the Points

After plotting the points, connect them to draw the rotated conic section. Ensure to follow the order of the points as determined by increasing \(\theta\).

Step 5: Complete the graph

Having connected the points, evaluate the graph. It should represent a conic section like an ellipse, parabola or hyperbola rotated by an angle of \(\pi /4\).