Problem 28 - Algebra and Trigonometry

Question

Select the representations that do not change the location of the given point. $$\left(4,120^{\circ}\right)$$ a. \(\left(-4,300^{\circ}\right)\) b. \(\left(-4,-240^{\circ}\right)\) c. \(\left(4,-240^{\circ}\right)\) d. \(\left(4,480^{\circ}\right)\)

Step-by-Step Solution

Answer

The representations that do not change the location of the given point \((4,120^\circ)\) are options c and d.

Step 1: Analyze option a

The coordinate is \(-4, 300^\circ\). A negative 'r' value means the point is reflected. Thus, this can be translated into a positive 'r' value with an angle of \(300^\circ + 180^\circ = 480^\circ\). That is not the same as the original given point so this option is not a valid representation.

Step 2: Analyze option b

The coordinate is \(-4, -240^\circ\). Reflecting the point because 'r' is negative turns this into a positive 'r' value with an angle of \(-240^\circ + 180^\circ = -60^\circ\). Adding multiples of 360° to make the angle positive, the new representation becomes \(4, 300^\circ\), which is not the same as the original coordinate. Therefore, this option is not a valid representation.

Step 3: Analyze option c

The coordinate given is \(4, -240^\circ\). Here, the 'r' value is positive and therefore there's no reflection needed. Converting the negative angle to a positive angle by adding 360°, the new representation comes out as \(4, 120^\circ\), which matches the original coordinate exactly. Hence, this option is a valid representation.

Step 4: Analyze option d

The coordinate is \(4, 480^\circ\). In this case, 'r' is positive, so no reflection is needed. Subtracting 360° from the angle due to the property of circularity, the new angle is \(480^\circ - 360^\circ = 120^\circ\). Hence, the new coordinate is \(4,120^\circ\), which is the same as the original coordinate. Thus, this option is also a valid representation.