Trigonometry and Turtle Graphics


This page is a guide to understanding the Trigonometry and Turtles Homework. Work it, own it, master it. SOH, CAH, TOA.

  1. What is the heading of the turtle with WHO number 2?.

    1. The who numbers go from 0 to 11, clockwise, starting from the turtle that faces straight north, with heading of 0 degrees.

    2. It can easily be seen that there are 3 slices of pizza pie or pumpkin pie in the upper right quadrant or quarter of the NetLogo grid. Since this is 90 degrees of the entire 360, it is clear that each slice must be 30 degrees.

    3. Or you can divide 360 by 12, which gives you the degrees of the angle between each pair of lines they have drawn.
      Whatever 
               cro n              cro n = create n ordered turtles
                     value is, 
                as in cro 32 or  
                      cro 8 or 
                      cro 36 or 
                      cro 72, 
                      
                the size of each slice is just 360 divided by n.
                
                360             360          360
               ----- = 10      ----- = 5    ----- = 30
                 36              72           12
                 
      

  2. What is the xcor value for the indicated turtle? It will be the same as the length of the side of the triangle that is adjacent to the 30 degree angle. Which side? The leg, not the hypotenuse. Every right triangle has two legs and one hypotenuse.

  3. What is the ycor for the indicated turtle? That will be the same as the length of the side opposite the 30 degree angle. Of course, we know all of the turtles started out at (xcor, ycor) = (0, 0).

  4. Always, always, always break down your problem into answering the following three questions:
      i.  What is known?  Write down what is given.
          Write down and determine what facts are 
          directly given or that can be derived from
          what is stated or seen in the problem diagram.  
          Note:  It can be VERY IMPORTANT for you to draw a diagram or picture of the situation.  
          It is NOT always given.
          Or you might add to the picture that is given, 
          like I do below here for you (Photoshop).
    
          This is your ammunition for trying to hit the target.
          Your bow and arrows. 
    
          Or this is your fuel for taking you from the starting point
          to the destination, from the given to the goal.
          
      ii. What is unknown?  What is the goal?
          What is the result that is desired?
          What is the question asking for?
          That is your target!
          That is your destination! 
    
     iii. How can you get from i. (given) to ii. (goal).
    
          You now have a clear idea of where you are at,
          and of where you want to go.
    
          What formulas or past similar problem and trips
          can help you get from i. to ii.?
    
          You might need the SINE or COSINE or TANGENT.
          Or for Monte Carlo you might need the 
                        b
                   a = --- d    formula.
                        c  
                                          rise
          Or you might need the slope = --------
                                           run
      
              formula where rise = (y2 - y1)
                        and run  = (x2 - x1)
    
              etc. etc. etc.
    
    
          Mobilize knowledge about circles, slopes,
          right triangles, points, distances between
          points, trigonometry that is relevant to
          the current problem and the i. given facts
          that you have WRITTEN DOWN and isolated and 
          the ii. relevant GOAL that you have also 
          WRITTEN DOWN as your clear target.      
    

  5. The three steps above are represented by the three Ghostbusters chararacters:
            i.  WHAT is given?        Dr. Peter Venkman
                WHAT is the goal?
                
           ii.  HOW to get from       Dr. Raymond Stantz
                    i. given start to 
                   ii. the desired result or goal.
                   
          iii.  Solve the problem using whatever tools
                (Netlogo, Vensim, calculator) and algebra
                you need to do.     
                                       Dr. Egon Spengler
    
  6. What is the solution for the xcor?
    cosine(30) = adj/hyp = adj/10
       
    Where does the 10 come from?  The turtles each went forward 10 units! (fd 10)
       
    cosine(30) using the calculator calc.exe 
    show cosine of 30 degrees is 0.866.
      
      0.866 = adj / 10   so multiply both sides
                         of the equation by 10,
                         we have:
                         
      (10)(0.866) = (10) (adj / 10) and 8.66 = adj, so the xcor answer is 8.66.
        
        Remember to SHOW your WORK!!!!    I am showing my work here, not just the final answer.
                           
    
  7. What is the solution for the ycor? SINcE ycor involves the side OPPOSITE to the 30 degree angle, we need to use the SINE.
    Sine(30) = OPP/HYP = OPP/10
      
    Sine(30) = 0.5 = OPPOSITE / 10 
      
               and 10(0.5) = OPPOSITE = 5.0.
               
    So the ycor is 5.0.
    
    The (xcor, ycor) for the indicated turtle 
         with who number 2
              is (8.66, 5.0)
    
  8. Is there another way to find the ycor, given that we know the xcor, i.e. we know the length of the ADJACENT LEG to the 30 degree angle?

    Yes. Knowing the adjacent leg has length 8.66 units and knowing the angle is 30 degrees and having as our goal what is the length of the OPPosite Leg gives the following setup:

    
          GIVEN                                       GOAL
          INPUT                                       OUTPUT
            i.                                          ii.
       Known or given:                      Goal, unknown to find, desired result:
       --------------                       -------------------------------------
           Angle 
              30 degrees
                                                Length of OPPosite side,
           Adjacent side length                    opposite leg of the 
              8.66 units                           triangle, side opposite 
                                                     the 30 degree angle.
    
          How do you get from i. to ii. from given to goal?
    
          TOA = Tangent Opposite Adjacent
    
                       Opposite        opp     O
          Tangent = --------------- = ----- = ---   Remember by thinking of
                       Adjacent        adj     A          stubbing your TOA...
    
          So the TOA formula is suggested when you focus on the fact
          that you know the 30 degrees and can get the T Tangent, thus
          with a calculator know the Tangent of 30 degrees, 
    
          and you know the length of the adjacent leg to the 30 degree 
          angle, which is 8.66.
                                opposite
          Tan(30) = 0.57735 = ------------ so 8.66 * 0.57735 = opposite length
                                  8.66     
                                          and 4.99985 - opposite length
    
                                         thus 5.0 = ycor