Problem 27 - Algebra and Trigonometry

Question

Select the representations that do not change the location of the given point. $$\left(7,140^{\circ}\right)$$ a. \(\left(-7,320^{\circ}\right)\) b. \(\left(-7,-40^{\circ}\right)\) c. \(\left(-7,220^{\circ}\right)\) d. \(\left(7,-220^{\circ}\right)\)

Step-by-Step Solution

Answer

The representations that do not change the location of the point (7,140°) are a. \(-7,320^{\circ}\) and b. \(-7,-40^{\circ}\).

Step 1 - Analysis and Comparison

Firstly, acknowledge that a change in sign from positive to negative for the radius (r) implies a 180° change in the angle. So, for the point (7,140°), equivalent representations with a negative radius could be (-7,320°) and (-7,-40°). An angle of 320° is equivalent to 140° + 180°, and an angle of -40° is equivalent to 140° - 180°. Compare this principle with the options given.

Step 2 - Option A

For option a, \(-7,320^{\circ}\), it's seen that the radius is -7 and the angle is 320°. This representation is equivalent to our original point (7,140°) because 320° is equivalent to 140° + 180°. So, option a does not change the location of the point.

Step 3 - Option B

For option b, \(-7,-40^{\circ}\), the radius is -7 and the angle is -40°. This representation is equivalent to our original point (7,140°) because -40° is equivalent to 140° - 180°. Therefore, option b does not change the location of the point.

Step 4 - Option C and D

For option c, \(-7,220^{\circ}\), and option d, \(7,-220^{\circ}\), both these representations change the location of the original point (7,140°). Hence, options c and d do change the location of the point and are not correct selections.