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Help with the unit circle please.

  1. Mar 25, 2006 #1
    Hi there.
    I'm really confused with this question, it's about the unit circle.
    Firstly I was asked to look up the values of (sorry I don't know how to put in the symbol for PI-3.14) 'cos(PI/3)' and 'sin(PI/3)' ok I got them.
    Then I'm told to use the unit circle diagrams (diagrams?) to show the connection between:
    cos(2PI/3), sin(2PI/3) and cos(PI/3), sin(PI/3)
    (Then again for cos(4PI/3), sin(4PI/3) and cos(PI/3), sin(PI/3)

    What is confusing me is the 'sin' and 'cos'! I don't understand where they fit into the unit circle?? I can find 2PI/3 on the unit circle, but how do I know if it is for sin or cos??

    And by connection, I firstly thought, 'Ok, PI/3 and 2PI/3 are basically both 60degrees from the positive y axis' but then that isn't right because I haven't considered cos or sin!

    If someone could point me in the right direction, it would be a huge relief!
     
  2. jcsd
  3. Mar 25, 2006 #2
    Ok, I don't know if there is some light shining through my mind...but I checked on the calculator some things.
    I put the mode on 'degree' and found out that cos60 (60degrees being that PI/3) is 0.5 and sin60=0.866.
    Now the 2PI/3 is 120degrees. I put cos120=-0.5 and sin120=0.866
    And for the 4PI/3-240degrees. Cos240=-0.5 and sin240=-0.866

    Of course the pattern or similarities are obvious...cos60/120 are the same, just in different quadrants...sin60/120 are exactly the same, sin60/240 are the same, just in different quadrants, and same for cos240.

    I'm still confused...and I feel like something is trying to get through to me...I'm searching around, but still I can't figure this out?? The question says to use unit circle to show the relationship...so does that mean I need to draw a unit circle and draw the angles for this all on it?

    I don't know:\
     
  4. Mar 25, 2006 #3
    The unit circle is such that the y-co-ordinate of an angle is the sine of that angle and the x-co-ordinate is the cosine of the angle. Thus, when you draw an angle, the mark the point where it intersects the unit circle. The y-co-ord is the sin and the x-co-ord is the cos.

    This is because for any point, y = r*sin(theta) and x = r*cos(theta). For an point on the unit circle, r = 1.
     
  5. Mar 25, 2006 #4
    forgive me if what i will be explaining you already know. After reading you message i believe you dont understand what sin and cos are and somewhat of the unit circle.

    A unit circle is a circle drawn on a Cartesian Plane. Such circle has a radius of 1. Your angle measures have been given in radians, if you feel more comfortable in degrees just as a quick reference you can change those measures by multiplying the radians x (180/pi).

    Anyways, imagine that you wrap a "number line" around the circle. Well, they give you an angle measure and you want to know where the end of line (of the angle) lies on the plane. Points on the Cartesian Plane are given as (x,y). Well, cosine would be the x, and sine would be the y. Lets get some easy ones if they ask you for example what is the cosine of pi/2, well (pi/2) x (180/pi)=90 degrees.... as i said the unit circle is a circle with a radius of one. So, that means that the endpoints of that line will be (0,1) since the question is asking for cosine(which is x) then cos= 0.

    Now, let me help you with that 2pi/3....I will convert it to degrees if it helps you, however, you should learn the basic measures in radians of the unit circle. ok, (2pi/3) x (180/pi)= 120 degrees, which will be on the second quadrant. Well, if you have a Ti calculator, you should change the mode(if you are going to be pluging in angle measures in degrees, have it on degrees. If you are going to plug in measures in radians put it in radians) now if simply you type cos(2pi/3) or cos(120) you should get the cosine, meaning the x of that line of the angle. cos of 2pi/3 is -.5, and the sine is .866 or square root of 3/2.... the cosine (x) is negative because the point lies on the 2nd quadrant.

    Hopefully i was able to help you out and hopefully you will understand the unit circle.

    p.s you should learn the sine and cos for the basic radians angle measures which are: 0, pi/6,pi/4,pi/3,pi/2,2pi/3,3pi/4,5pi/6,pi,7pi/6,5pi/4,4pi/3,3pi/2,5pi/3,7pi/4,11pi/6
     
  6. Mar 25, 2006 #5
    Thank you both of you! Some things were made clear.

    So, I already have that -.5 and .866 etc...do I plot them on the unit circle and join the points?? Then I should see the connection between them?? I think all of them in the end will have the same angle??
     
  7. Mar 26, 2006 #6
    Ok I think I get that now!

    I have another question, where I need to use trig formula's.
    It says to expand this term and simplify:
    cos(A)+cos(A+2PI/3) + cos(A+4PI/3)
    I'm not sure where to stay with this one? Should I put together 'cos(A+2PI/3) + cos(A+4PI/3)'? And then simplify the 4PI/3 to get 2PI/3 so that i can get 1 for that part and then it should equal cos(A)??
     
  8. Mar 26, 2006 #7
    I think you have to use the trigonometric expansion formulas for cos(a+b) ... cos(a+b) = cosa.cosb - sina.sinb
     
  9. Mar 27, 2006 #8
    Yeah that's it! I figured it out:) Thanks!
     
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