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Radian Conversion

Date: 07/24/2012 at 11:00:45
From: Niras
Subject: sin in radians

I need -- NEED -- to brush up my skills in trigonometry, seriously.

In calculating trig values, the sin of 30 degrees is .5.

But with the calculator set to radians, sin(30) is now -0.988031624.

Can you PLEASE show me the steps as to how?

I know how to work with degrees, but tried to stay away from radians. Now
it is inevitable, and I do not know how to do much calculating involving
radians. But with your help, I believe I will eventually conquer it

Date: 07/24/2012 at 11:20:35
From: Doctor Ian
Subject: Re: sin in radians

Hi Niras,

You can think of degrees and radians as being like miles and kilometers.

For example, if you are traveling 30 miles per hour, that's a different
speed than 30 kilometers per hour -- even though the numbers are the 
same -- right?

More to the point, suppose you have a circular track that is 2 miles
around. If you start at point X and drive 30 miles, you end up where you
started; you make 15 complete trips.

But if you drive 30 kilometers, where will you end up?

A kilometer is about 5/8 of a mile, so

           5/8 mi
   30 km * ------ = 150/8 mi, or 18 3/4 miles.  
             1 km

So that would be 9 complete trips around, plus another 3/4 miles, or 3/8
of the way around the track.

Make sense?

In the same way, suppose you have an angle with a measure of 30 radians.
Two pi radians takes you all the way around a circle (because the
circumference is equal to twice pi times the radius). So two radians is
equivalent to 360 degrees. That means

            360 deg
   30 rad * ------- = 1718.87 degrees
            2pi rad

That's four times around the circle, plus 278.87 degrees. So

   sin(30 rad) = sin(278.87 deg) = -0.988

Generally, when working with radians, it helps to keep a few key values in

   360 degrees = 2pi   radians
   180 degrees =  pi   radians
    90 degrees =  pi/2 radians
    60 degres  =  pi/3 radians
    45 degrees =  pi/4 radians
    30 degrees =  pi/6 radians

You can get those last four by dividing 180 degrees into 2, 3, 4, and 6

In the same way, you could figure out that 36 degrees is pi/5, 15 degrees
is pi/12, and so on. That's one of the nice things about 180 -- it has
lots of divisors -- so you can make lots of exact fractions with it. 

Does this help? 
- Doctor Ian, The Math Forum 

Date: 07/24/2012 at 12:00:20
From: Niras
Subject: Thank you (sin in radians)

Doctor Ian!

Got it!

Thank you!
Associated Topics:
High School Trigonometry

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