Problem 29 - Algebra and Trigonometry

Question

Select the representations that do not change the location of the given point. $$\left(2,-\frac{3 \pi}{4}\right)$$ a. \(\left(2,-\frac{7 \pi}{4}\right)\) b. \(\left(2, \frac{5 \pi}{4}\right)\) c. \(\left(-2,-\frac{\pi}{4}\right)\) d. \(\left(-2,-\frac{7 \pi}{4}\right)\)

Step-by-Step Solution

Answer

Among the provided options, only option b, i.e., \((2, 5\pi/4)\) doesn't change the location of the given point.

Step 1: Understand Polar Coordinates Representation

Polar coordinates \((r, \theta)\) can represent a location in a two-dimensional plane. Here, \(r\) represents the distance of a point from the origin, and \(\theta\) is the counterclockwise angle from the x-axis. The same point can have many equivalent polar representations. Adding or subtracting a multiple of \(2\pi\) to/from the angle doesn't change the point location because it results in a full rotation around the origin.

Step 2: Compare the Polar Coordinates

Now, compare the given point \((2, -3\pi/4)\) with the options. For representation to be equivalent, the distance \(r\) should be the same, and the angles should differ by a multiple of \(2\pi\).

Step 3: Analyze Option (a)

Option a: \((2, -7\pi/4)\). Here, \(r\) is the same as the given point. The difference of angles is \(-7\pi/4 - -3\pi/4 = -\pi\), which is not a multiple of \(2\pi\). Hence, this point is not equivalent to the given point.

Step 4: Analyze Option (b)

Option b: \((2, 5\pi/4)\). Here, also \(r\) is the same. The difference of angles is \(5\pi/4 - -3\pi/4 = 2\pi\), which is a multiple of \(2\pi\). Hence, this point is equivalent to the given point.

Step 5: Analyze Option (c)

Option c: \(-2, -\pi/4)\). Here, \(r\) differs from the given point. Hence, this point is not equivalent to the given point.

Step 6: Analyze Option (d)

Option d: \(-2, -7\pi/4)\). Here also, \(r\) differs from the given point. Hence, this point is not equivalent to the given point.