Question

Using complete sentences, define and compare radian measure to degree measure. In doing so, be sure to answer each of the following questions:

•When would degree measure be appropriate?
•When would radians be a better choice?
•What are the pros and cons of each?
2.Adam and Beth are visiting an amusement park and have decided to ride the carousel. Adam picked a horse on the outside edge and Beth chose a dragon on the inside, closer to the center.

Part 1: Do Adam and Beth travel the same distance during the ride? Choose a distance that each seat (horse and dragon) sits from the center and use the radius to determine how far each would travel during one rotation.

Part 2: Choose an angle of rotation. Using complete sentences, compare the distance Adam and Beth will travel during this angle measurement.

Part 3: Using complete sentences, describe which position you would prefer and why.
3.Robbie the Robot is on a weather satellite orbiting Earth about 3600 km above the surface. The Earth’s radius is about 6400 km. He has had a malfunction in his output device, and the satellite is traveling without communication. His last report was only in terms of trigonometric values and was only partially received. It said, sin Θ < 0….. and then he was lost again.

Part 1: Create a set of coordinates that would be reasonable for Robbie’s position in space and satisfy his last, partial report. Using complete sentences, describe Robbie’s location and your reasoning.

Part 2: What are the values of the sin Θ, cos Θ, and tan Θ using your coordinate point?

Answer 1

For question 1:  Consider a circle with centre A and radius r.  Let B and C be points on the perimeter, such that the distance from B to C (measured along the perimeter) is also r.  Angle BAC is then defined as one radian, i.e. a radian is that angle at the centre of a circle which subtends an arc length equal to the radius of the circle; put another way: arc length equals radius times number of radians at the centre.  Now, if we move point C along the perimeter away from B until it reaches B, i.e. so that the distance from B to C measured along the perimeter equals the circumference of the circle, then let the angle BAC which subtends the arc length equal theta.  According to the definition of a radian, arc length (which equals circumference) = r times theta (with theta the radian equivalent of 360 degrees).  We know from calculus (but that is a different story, so just accept it for now) that the circumference of a circle equals 2 pi r.  Therefore,  2 pi r  =  theta times r, which means theta equals 2 pi radians.  Because pi = 3,141 592 65 .... (also from calculus),  360 degrees equals 2 pi radians which equals 3,141 592 65 ... radians, or 1 radian equals approximately 57,295 77 ... degrees.  Hope it helps.