How do you find the reference angle for (theta) = (6pi) / 7?

2 Answers
Apr 25, 2015

Okay, first of all, the problem lies in the format we can see the angle in. It seems to be in radians as it is using measures of pi, so here's what we can do:

There's a short way to do it and a long way to do it; I'll explain both.

Let's say we were converting an angle of 50° to radians, using measures of pi:

If at any point you want another example on doing this, check here.

I'll start with the long version first:

So you can use this little formula:

x° " in "Radians = (x * pi)/180 where is the angle you are converting (in this example, 50°)

Your case is stated after I explain this method forwards.

All you have to do is substitute it in and then simplify:

x° " in "Radians = (50 * pi)/180

x° " in "Radians = (5 * pi)/18

x° " in "Radians = 15.708/18 (Rounding numbers to 3 DP )

x° " in "Radians = 0.873 Radians (Rounding numbers to 3 DP )

So we could do the same process to go from radians to degrees:

x° " in "Radians = (x * pi)/180 where is the angle you are finding.

All we have to do is substitute in the number of radians we want to convert (in this case we'll use the answer we got in our last conversion [0.087])

In your case, to get the number of radians, just simplify what you have e.g (6 * pi)/7 = 18.85/7 (to 3DP) and so on...

0.873 = (x * pi)/180

Then manipulate it to find x:

0.873 * 180 = x * pi

157.14 = x * pi

157.14/pi = x

50.029° = x

x = 50.029° (Answer is slightly out as we rounded our answers half way through)

The shorter method may seem more simple:

We know that for every pi radians , there is 180°, or:

pi Radians = 180°

So, in your case, we have (6pi)/7 radians, which is the same as 6/7pi.

Now, if pi radians is 180°, all we have to do is change it:

6/7pi = (6/7)180°

Now simply multiply it in whichever method you wish:

(6/7)180 = 154.286°

Now that we have dealt with the format and put it into degrees - it's easy to use:

To figure out which quadrant it is in (i.e. which formula to use), we need to imagine the quadrants:

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When we figure out which quadrant our angle (154.286°) is in, we can decide which formula to use.

As 150.286° is between 90° and 180°, it is in the second quadrant , where we use 180° - x° (as we use the x axis as the point of reference).

Finally,

Ref angle = 180° - 150.286°

Ref angle = 29.714°

Apr 25, 2015

pi = (7pi)/7 (I'll assume you know what pi looks like in standard position.

(6pi)/7 falls short by pi/7

The reference angle is pi/7