Degrees and Radians, ExplainedDate: 03/19/2002 at 23:09:20 From: Mandy Subject: Degrees and radians How do you find the degree measure for an angle from pi/60 rad? I have a chart that I made of conversions, but I don't know how to do it when pi is added. If the radian measure of an angle is doubled, is the degree measure of the same angle also doubled? Thank you. Date: 03/20/2002 at 09:08:00 From: Doctor Peterson Subject: Re: Degrees and radians Hi, Mandy. Radians and degrees are just different units for measuring angles, like feet and meters; so they are proportional, and when you double an angle, its measure doubles whether in degrees or radians. Imagine drawing an angle, and then drawing a circle (the size doesn't matter) with its center at the vertex. We measure an angle by measuring the length of arc subtended by the angle - that is, by finding how long the part of the circle between the two sides of the angle is. (That's what a protractor does, for example; it's just a ruler curved around a circle.) To measure in degrees, we might wrap a flexible measuring tape around the whole circle and mark off the circumference; then lay it out straight and divide that into 360 equal parts, so that each degree on the tape is 1/360 of the whole circumference. Now we can wrap it back around the circle, and measure the angle by counting degrees. To measure in radians, we lay the tape straight and mark off the radius of the circle. That itself becomes our unit - do you see how much more natural that is than degrees? We can then divide the radian into whatever parts we choose, say hundredths. Then we wrap it around the circle and measure how many radians, in hundredths, the angle covers. If you think about this, you realize that a whole circle will be 2 pi, or about 6.28, radians, since the circumference is 2 pi times the radius, which is the unit we used. Since a whole circle is also 360 degrees, just because that is the number of parts we chose to divide the circle into, we see that 360 degrees = 2 pi radians Now we can convert between the two units by using the fraction 360 degrees ------------ = 1 2 pi radians For example, given an angle of pi/60 rad, this is equal to 360 deg pi 360 pi/60 rad * -------- = -- * ---- deg = 3 deg 2 pi rad 60 2 pi because we can cancel pi, 60, and 2 from the product. It is also important to know that the area of a sector is proportional to the angle, so that for example a 90 degree sector is 90/360 = 1/4 of a circle, and has area 1/4 pi r^2. Write back if you need more help. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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