Pythagoras and Tangent

P and T Pythagoras and Tangent - Where is Waldo

This page is a guide to understanding the TOA solution to 11 Turtles and Tangent April 7th, 2016 example.
Work it, own it, master it. SOH, CAH, TOA.

What is the heading of the turtle with WHO number 2?.
1. The who numbers go from 0 to 10, clockwise, starting from the turtle that faces straight north, with heading of 0 degrees.
2. Divide 360 by 11, which gives you the degrees of the angle between each pair of lines they have drawn.
```
          
          360             
         ----- = 32.727 degrees per slice (if we were slicing pizza or pie
           11                                 into 11 equally sized slices).
           
```
3. 32.727 + 32.727 = 65.454 degrees is the HEADING of the turtle with WHO #2.
4. The angle of interest in the right triangle drawn above is then:
```
       90
     - 65.454
    ---------
       24.546 degrees is the angle shown for the right triangle.
```
What is the xcor value for the indicated turtle?
It will be the same as the length of the side that is adjacent to the 24.546 degree angle. SOLVE IT USING CAH.
Which side? The leg, not the hypotenuse.
Every right triangle has two legs and one hypotenuse.
What is the ycor for the indicated turtle?
That will be the same as the length of the side opposite the 24.546 degree angle. Solve it using SOH.
Of course, we know all of the turtles started out at (xcor, ycor) = (0, 0).

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.

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

What is the solution for the xcor? (Using only the COSINE and CAH)

cosine(24.546) = adj/hyp = adj/10
   
Where does the 10 come from?  The turtles each went forward 10 units! (fd 5 + fd 5)
   
cosine(24.546) using the calculator calc.exe 
show cosine of 24.546 degrees is 0.9096.
  
  0.9096 = adj / 10   so multiply both sides
                      of the equation by 10,
                      we have:
                     
  (10)(0.9096) = (10) (adj / 10) and 9.096 = adj, so the xcor answer is 9.096.
    
   Remember to SHOW your WORK!!!!    I am showing my work here, not just the final answer.

What is the solution for the ycor?
SINcE ycor involves the side OPPOSITE to the 24.456 degree angle, we need to use the SINE. SOH.

Sine(24.456) = OPP/HYP = OPP/10
  
Sine(24.456) = 0.414 = OPPOSITE / 10 
  
           and 10(0.414) = OPPOSITE = 4.14.
           
So the ycor is 4.14.

The (xcor, ycor) for the indicated turtle with who number 2 is (9.096, 4.14)

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 24.456 degree angle?

Yes. Knowing the adjacent leg has length 9.096 units and knowing the angle is 24.456 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 
          24.456 degrees
                                            Length of OPPosite side,
       Adjacent side length                    opposite leg of the 
          9.096 units                           triangle, side opposite 
                                                 the 24.456 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(24.456) = 0.455 = ------------ so 0.455 * 9.096 = opposite length
                              9.096     
                                      and 4.136 = 4.14 = opposite length
                                                                     thus 4.14 = ycor

Solving using ONLY THE TANGENT given the starting point information of the 24.456 degree angle and length 10 hypotenuse.

    2     2     2
   a  +  b  =  c  is the pytagorean theorem

    2     2     2                                            opposite      ycor
   a  +  b  = 10  = 100   TANGENT(24.456 degrees) = 0.455 = ----------- = ------
                                                             adjacent      xcor
            ycor
   0.455 = ------ so ycor = (0.455)(xcor)
            xcor

        2        2
    xcor  +  ycor  = 100

        2              2
    xcor  + (0.455xcor)  = 100 

        2         2         2                                            2
    xcor  + 0.207x  = 1.207x  = 100 and after taking square root ( 1.207x ) 
                                                          we have 1.0986x = 10,
                                10
                         and -------- = 9.10 which is the xcor and length of adj leg.
                              1.0986

    The opposite side is 0.455x = (0.455)(9.10) = 4.14 units is the opposite leg,
                                                             which is the ycor.