What Is a Principal Angle and How Is It Calculated?

[aisha]

Can someone please explain to me what Principal angle is, all i have is a one sentence definition but I don't understand what to do or how to find the answer.

The angle between 00 and 3600 is called the principal angle

Find the principal angle of the followings.

5000
- 1500
12500
1350

I have no clue I am new to this.

 

[quasar987]

Between 00 and 3600 ?! They probably meant "between 0° and 360°".

In this case, it helps to visualise the problem by imagining a circle and an arrow or something indicating the angle*. Then they want to know what angle between 0° and 360° does the arrow shows after it has rotated 5000°. Dividing 5000 by 360 tells us how many complete circle the arrow has made. 5000/360 = 13+(125/9). It has made 13 complete circles and 125/9 th of one. Then, (125/9)*360° tells us how many degrees does 125/9 th of a circle corresponds to. It is 320°.

*Conventionally, if you draw a cartesian coordinate system through your circle with the origin at the center, 0° correspond to the line given by the x-axis and a positive angle increment means rotating counter-clockwise.

 

[aisha]

Ok I understand sort of, I guess I start at the initial arm in standard position sorry the first question was 500 degrees not 5000 but anyways no big deal. I understand how 5000 degrees made 13 complete circles but the 125/9 is confusing me, I don't know how to get the degrees from that (125/9)*360 degrees gives me 5000 not 320 degrees how do I get 320 degrees and what do I do with this?

So far I went 13 times around the circle from the initial arm how much more do I go?

 

[dextercioby]

Ok I understand sort of, I guess I start at the initial arm in standard position sorry the first question was 500 degrees not 5000 but anyways no big deal. I understand how 5000 degrees made 13 complete circles but the 125/9 is confusing me

If 360°----------------->one rotation,then
5000°----------------> 'x' rotations

Then $$x=\frac{5000}{360}=\frac{125}{9}=13\frac{8}{9} rotations$$
That fraction of $\frac{8}{9} rotations$ can be computed via the same technique

If 360°----------------->one rotation,then
'y'°------------------>8/9 rotations

U find 'y'=320°.

I don't know how to get the degrees from that (125/9)*360 degrees gives me 500 not 320 degrees how do I get 320 degrees and what do I do with this?

So far I went 13 times around the circle from the initial arm how much more do I go?

320° degrees?That's the angle the problem's asking you.The principal angle of 5000° is 320°.
Follow the same pattern for the other 4 angles.

Daniel.

 

[quasar987]

Ok I understand sort of, I guess I start at the initial arm in standard position sorry the first question was 500 degrees not 5000 but anyways no big deal. I understand how 5000 degrees made 13 complete circles but the 125/9 is confusing me, I don't know how to get the degrees from that (125/9)*360 degrees gives me 500 not 320 degrees how do I get 320 degrees and what do I do with this?

So far I went 13 times around the circle from the initial arm how much more do I go?

Oh yes, I made a mistake in typing the solution. Here's what it should have been:

"In this case, it helps to visualise the problem by imagining a circle and an arrow or something indicating the angle*. Then they want to know what angle between 0° and 360° does the arrow shows after it has rotated 5000°. Dividing 5000 by 360 tells us how many complete circle the arrow has made. 5000/360 = 125/9=13+(8/9). It has made 13 complete circles and 8/9 th of one. Then, (8/9)*360° tells us how many degrees does 8/9 th of a circle corresponds to. It is 320°."

Sorry for confusing you.

 

[quasar987]

Try it for the angle of 500° and show us what you did if you don't get the right answer. The answer is 140°.

 

[aisha]

ok I got the answer 140 degrees

doing it the same way as dextercioby, now I am confused on how to sketch this in a circle.

I have a cartesian plane going through a circle. For 140 degrees I started with the initial arm in standard position and the terminal arm is in the second quadrant.

$$x=500/360=25/18=1\frac{7} {18}$$

Since it is $$1\frac {7} {18}$$ then when sketching should I go 1 whole rotation plus the 7/18=140 degrees I know that equals 500 degrees again, taking us back to the main question, but I am trying to understand why did we only look at the fraction and not the 1 whole rotation? If it wasnt one would we still only look at the fraction rotation? Basically now I need some pointers on how to sketch these principal angles, please help thanks

wouldnt it have been easier to just do 500 degrees subtract 360?

 

[quasar987]

Since it is $$1\frac {7} {18}$$ then when sketching should I go 1 whole rotation plus the 7/18=140 degrees?

Try it. You will see that going one whole rotation plus 140° degrees gets you to the same point as just going 140°. But the answer to your question is 'No'. That is not what is asked of you in the question "Sketch the principal angle" because by definition, the Principal angle is 140°. So don't draw 500°, draw only 140°.

I know that equals 500 degrees again, taking us back to the main question, but I am trying to understand why did we only look at the fraction and not the 1 whole rotation? If it wasnt one would we still only look at the fraction rotation?

Yes, that 'fraction rotation' is the definition of the Principal angle.

 

[quasar987]

wouldnt it have been easier to just do 500 degrees subtract 360?

That way works too!

In general, it you want to find the Principal angle of N°, just subtract N° - 360° - 360° - ... - 360° until what you have left is a number between 0° and 360°. Then that is your principal angle.

 

[aisha]

ok so if we had 1000 degrees then we would have to just subtract 360 until we got a number that was between 0 degrees and 360 degrees?

That sounds easy if that's correct, but back to my last post and sketching please

 

[dextercioby]

why did we only look at the fraction and not the 1 whole rotation? If it wasnt one would we still only look at the fraction rotation? Basically now I need some pointers on how to sketch these principal angles, please help thanks

Okay,you're asked for an angle that should be between 0 and 360° which is less than a rotation okay??(The case 360° is noninteresting,as it is the same with 0°).So the integer number of rotations (the one that is at the left of the fraction line) is not interesting since it is a multiple of 360°,and therefore is equivalent with 0 rotations (or with the 0° angle).
In case of the negative angles,u must make that rotation of the arm clockwise,while in the case of positive angles,u'll have to make it anticlockwise (trigonometric sense).Any integer number of rotations (no matter the sense) is equivalent to 0° so therefore it's not relevant.
For angles in the negative interval (-360°,0°),u must find the principal angle,which must be between 0° and 360°.The transformation is achived simply by adding 360°.

Example:Consider the angle of -4696°.Divide it by 360°.U'll get:
1 rotation anticlockwise -------------------> +360°
1 rotation clockwise------------------------>-360°
x rotations clockwise------------------------>-4696°

$$x=\frac{-4696}{-360}=13\frac{16}{360}$$ rotations clockwise.
Again,neglect the integer number of rotations (no matter the sense) a,d concentrate upon the fraction.
1 rotation clockwise-------------------->-360°
16/360 rotations clockwise--------------> y°

U find y=-16°.However,u need the positive angle correseponding to -16°.And that is
$$Pos.ang.=-16+360=+344$$
,which is in the interval [0,360°].


Daniel.

 

[plusaf]

principal angles

ok I got the answer 140 degrees

doing it the same way as dextercioby, now I am confused on how to sketch this in a circle.

I have a cartesian plane going through a circle. For 140 degrees I started with the initial arm in standard position and the terminal arm is in the second quadrant.

$$x=500/360=25/18=1\frac{7} {18}$$

Since it is $$1\frac {7} {18}$$ then when sketching should I go 1 whole rotation plus the 7/18=140 degrees I know that equals 500 degrees again, taking us back to the main question, but I am trying to understand why did we only look at the fraction and not the 1 whole rotation? If it wasnt one would we still only look at the fraction rotation? Basically now I need some pointers on how to sketch these principal angles, please help thanks

wouldnt it have been easier to just do 500 degrees subtract 360?

yes, for 500 degrees, subtracting 360 is easy and pretty straightforward. unfortunately, for 5000 degrees, division is easier and faster than multiple subtraction...

i did some googling, and here's a link that tells a little about principal angles...
http://webalgebra.math.uiuc.edu/arbangls/rbai.htm

the basic deal is this: if you look at an angle, you're probably interested in some geometrical function of it, as sine, cosine, etc. the "problem" is, that you can't know how you got to that angle if it's less than 360 degrees! perhaps you went right from zero degrees to, say, 140 degrees; but you also could have gone 360+140, or 360+360+140, etc.

the bottom line is, that no matter how you got there, the trig functions of ("n" times 360 degrees plus X degrees) are exactly the same as the trig functions of "x degrees" alone!

therefore, "x" is referred to as the Principal Angle", and the fun is in finding out what the principal angle is if the "how you got there" is more than 360 degrees.

hope that helped, maybe a little...

 

[aisha]

ok ok ok Dextercioby thanks soo much lol I really do understand now, and really thanks i get it now totally.

Plusaf thanks for the website and thanks to everyone else who helped me out.