Problem 31 - Algebra and Trigonometry

Question

Select the representations that do not change the location of the given point. $$\left(-5,-\frac{\pi}{4}\right)$$ a. \(\left(-5, \frac{7 \pi}{4}\right)\) b. \(\left(5,-\frac{5 \pi}{4}\right)\) c. \(\left(-5, \frac{11 \pi}{4}\right)\) d. \(\left(5, \frac{\pi}{4}\right)\)

Step-by-Step Solution

Answer

The representations which do not change the location of the point \((-5, -\frac{\pi}{4})\) are options a and c.

Step 1: Identify the Given Point

The given point in polar coordinates is \((-5, -\frac{\pi}{4})\).

Step 2: Understanding Polar Coordinates

In polar coordinates, a point is represented by (r,θ). If we add or subtract multiple of \(2\pi\) to the θ, the location of the point remains unchanged. This is because the constant \(2\pi\) represents a full rotation around the origin. Thus, the position of the point remains the same after a full rotation.

Step 3: Check Each Representation

Now check each provided representation to see if they represent the same position as the given point: \n a. \((-5, \frac{7\pi}{4})\): Adding \(\frac{8\pi}{4}\) to the θ component of the given point gives \(-\frac{\pi}{4} + \frac{8\pi}{4} = \frac{7\pi}{4}\). So, option a represents the same point. \n b. \((5, -\frac{5\pi}{4})\): This option has changed the r component, and hence, represents a different point. \n c. \((-5, \frac{11\pi}{4})\): Adding \(\frac{12\pi}{4}\) to the θ component of the given point gives \(-\frac{\pi}{4} + \frac{12\pi}{4} = \frac{11\pi}{4}\). So, option c represents the same point. \n d. \((5, \frac{\pi}{4})\): This option not only changes the r component but also the θ, and hence, represents a different point.