Weird trig 4

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Weird trig 4

  1. 1. Weird Trigonometry (4) By Dave C, 2013
  2. 2. Sin(x) 2Cos(x) 2The ‘proper’ way to solve this is to combinethe two trig terms into a single trigterm, using the replacement... 2 2 1 BASin(x) BCos(x) A B Sin x Tan (A)In this case... 2 2 1Sin(x) 2Cos(x) 1 2 Sin x Tan (2)
  3. 3. Sin(x) 2Cos(x) 2That’s easier said than done. You have to remember thetransformation, or know where to look it up easily, or know howto reproduce it from basic principles.There’s an easier way.It’s an approximation that gets you close to the answer with verylittle effort.It works for numbers close to the middle of the triggraph, i.e., values close to 45 degrees.
  4. 4. Sin(x) 2Cos(x) 2 Notice how the Sine and Cosine graphs cross each other at the 45-degree mark.As one decreases to the left or right, the other increases by nearlythe same amount, so the sum of the two remains nearly constant.
  5. 5. Sin(x) 2Cos(x) 2 Sin(x) Cos(x) 2 ...for values close to 45OAs one decreases to the left or right, the other increases by nearlythe same amount, so the sum of the two remains nearly constant.
  6. 6. Sin(x) 2Cos(x) 2 Sin(x) Cos(x) 2 ...for values close to 45OThe deviation does not exceed 5% for values between 30-60O.
  7. 7. Sin(x) 2Cos(x) 2 So for this problem you could say... Sin(x) Cos(x) 2 Sin(x) Cos(x) Cos(x) 2 2 Cos(x) 2 ...which is a much easier problem to solve.
  8. 8. This gives a value of 54O.Sin(54) 2Cos(54) 1.985(True value 53.5 degrees)
  9. 9. [END]

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