Expert: Josh - 12/2/2009

Question
Hi Josh,

can you help me please,


Q1    change rads to degrees       0.6 rads = 34.38 degrees (2d.p.)
            is this correct as my paper says 15.3 degrees

I got the other questions in this section using the formula

          180/pi x amount of rads = degrees,
yet, i must be doing something wrong with Q1.



q2   A sector of a circle has area 50cm^2 with radius 5cm.What is the angle in radians? How do i get to this solution?I know about turning degrees in to rads and vice versa, but not quite sure how to go about it with an area.

answer       4 rads



many thanks
Richard


Answer
Hi Richard,

Q1. To convert from x degree to radian, compute (x/180)*pi [radian].
To convert from w radian to degree, compute (y/pi)*180 [degree].

The following is all I remember, no formula is really necessary.
180 degrees corresponds to pi radian, where pi is approximately 3.141592.
If you can remember this, you can do no wrong.

e.g., 90 degrees = pi/2 radian.
e.g., 60 degrees = (60/180)*pi = pi/3 radian.
e.g., 145 degrees = (145/180)*pi = 3*pi/4 radian.

Q2. Don't be bothered by "degree" or "radian" first up, we will deal with this latter. This is how I derive the formula for a sector. The area of a circle with radius "r" is A(circle)=pi*r^2. As you know, spinning a complete revolution about the center of the circle gives 360 degrees [or 2*pi radian]. So, the area for a sector with angle "x" (degree) can be expressed as a fraction of A(circle). In particular,

If angle is measured to be "x" degrees:
A(sector; x degree) = (x/360) *pi*r^2.
If angle is measured to be "w" radian:
A(sector; w radian) = w/(2*pi) *pi*r^2 = (w/2)*r^2.

OH NO! NOT ANOTHER FORMULA
==========================
Note: This result is not something worth committing to memory in my opinion. If remembering it works for you, then by all means remember it. But knowing the fundamentals (here, the formula for a circle, and seeing the area of a sector as a fraction of this) the necessary results can always be derived -- provided you can see the connection.

Using the second form, solving 50=(w/2)*(5^2) for angle w,
| 50=(w/2)*25,
| w=4 radian.
| This is equivalent to (4/pi)*180 = 229.18 degrees.

I remember you asking about partial fraction decomposition...have you got that sorted now? If anything is still unclear, let me know.