Question

The steering wheel on Becky’s car has a 16-inch diameter, and its center is at (0, 0). Point X at the top of the wheel has coordinates (0, 8). To turn left off her street, Becky must rotate the steering wheel by 300°. Find the coordinates of X when the steering wheel is rotated. Round to the nearest tenth.

Answer 1

Rotating the steering wheel left 300 degrees is the same as turning it right 60 degrees.  Well, not if you're driving, but the same as far as where the points on the wheel end up.

In polar coordinates X is P(8, 90°), which is how I'll write magnitude 8, angle 90 degrees.  In rectangular coordinates that's (8 cos 90°, 8 sin 90°) = (0,8) as we're told.

After rotation by 300° the angle of X is 90° + 300° = 390° so our coordinates are

P(8, 390°) = (8 cos 390°, 8 sin 390°) = (8 cos 30°, 8 sin 30°)

= (8 (√3/2), 8 (1/2)) = (4√3, 4)

Answer:  (4√3, 4)

I'll leave the rounding to you; I hate ruining a nice exact answer with an approximation.

As we see we would have reached the same 30° point if we had rotated -60° instead of 300°.

Answer 2

Answer:

Rotating the steering wheel left 300 degrees is the same as turning it right 60 degrees.  Well, not if you're driving, but the same as far as where the points on the wheel end up.

In polar coordinates X is P(8, 90°), which is how I'll write magnitude 8, angle 90 degrees.  In rectangular coordinates that's (8 cos 90°, 8 sin 90°) = (0,8) as we're told.

After rotation by 300° the angle of X is 90° + 300° = 390° so our coordinates are

P(8, 390°) = (8 cos 390°, 8 sin 390°) = (8 cos 30°, 8 sin 30°)

= (8 (√3/2), 8 (1/2)) = (4√3, 4)

Answer:  (4√3, 4)

I'll leave the rounding to you; I hate ruining a nice exact answer with an approximation.

As we see we would have reached the same 30° point if we had rotated -60° instead of 300°.