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Vraag 1


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  1. Use the cos-formula, which is a generalisation of the Theorem of Pythagoras!
  1. We can without loss of generality place PQRS at the origin, with coordinates as shown. So:
    LHS = PR2 + QS2
    = (b + a)2 + c2 + (b – a)2 + c2
    = 2(a2 + b2 + c2)

    Similarly:
    RHS = PQ2 + QR2 + RS2 + SP2
    = (b2 + c2) + a2 + (b2 + c2) + a2
    = 2(a2 + b2 + c2)
This shows how powerful an algebraic approach, using coordinates, can be in Geometry!

 

Vraag 3
  1. Kliek hier vir Excel oplossing.

  2. Kliek hier vir Excel oplossing.

    Note: Many student do not seem to understand the difference between, and the connections between the concepts domain, range, amplitude, period, phase and the concept of parameters, i.e. a, b, c and d in y = acos[b(x - c)] + d ...

    If we can calculate or estimate the parameters, it will make the numeric calculation of the best-fit equation much easier!

    a is the amplitude – e.g. for a simple harmonic motion, it is the maximum movement from rest. So a is half of the distance between the smallest and largest values:

    So 4,35cos b(x – c) lies between -4,35 and 4,35 for all b and c. But the temperature graph lies between 12,5 and 21,2. So it is d that moves the graph up.
    So: d = 21,2 – 12,5 = 12,5 – (-4,35) = 16,85

    b is the "speed" of movement, b determines how many cycles there are in one period.
    The period of cos x is 360°, of cos 2x is 180°, of cos 3x is 120° …
    So cos bx has a period of .
    But we are working in radians, and the period of this year temperature is obviously 12!
    So This will be the same for any city!

    y = cos (x – c) is the graph of cos x that moves c units right for c>0. But note that cos (2x – 60) does not move 60°, but 30°! So the graph of cos (bx – c) moves c/b units, but the graph of cos b(x – c) moves c units. So the form y = acos b(x .– c) is much more useful than y = acos (bx – c) …
    We cannot calculate c exactly from the data – at best we can say that maybe (2; 21,2) is a turning point of the graph, which will mean that the graph has moved 2 units to the right, then c = 2. But the turning point is probably between x = 1 and 2, so c is between 1 and 2.

Vraag 4
Ons verwag teoreties DRIE snypunte (hoekom?).Verstel die gebied en terrein (WINDOW), sodat jy 'n redelike deel van die grafiek kan sien, byvoorbeeld:

  

Gebruik nou vir elke snypunt ZOOM en CALC intersect om die koordinate van die snypunte te bepaal:
(-3,3684;-6,0526)
(-0,4787; -1,7365)
(1,8499; 1,7748)
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