Weird trig 1

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Weird trig 1

  1. 1. Weird Trigonometry (1) By Dave C, 2013
  2. 2. In this presentation I will work through a trigonometryproblem in an offbeat way.An answer will be found to a tricky-looking problem withoutthe use of a calculator (except at the end when I want tocompare the accuracy of my answer with what the calculatorwould give me).The point of it all is to show that trig isn’t always asconvoluted as it appears in the classroom. There areshortcuts you can use.
  3. 3. Cos(x) Sin(x) 0.9 Find x
  4. 4. Cos(x) Sin(x) 0.9First sketch a graph of these functions and find a placewhere the height difference is about 0.9.
  5. 5. Cos(x) Sin(x) 0.9This is the Sine graph:Basically a straight line followed by a curve to horizontal.
  6. 6. Cos(x) Sin(x) 0.9And here’s the Cosine graph:Just the same thing drawn backwards.
  7. 7. Cos(x) Sin(x) 0.9 0.9Looks like the height difference is 0.9 over here.
  8. 8. Cos(x) Sin(x) 0.9 0.1The Cosine line is nearly flat at this point.That means that the Sine value at this point is about 0.1.
  9. 9. Cos(x) Sin(x) 0.9 Sin(x) 0.1 I can use that information to make the problem simpler. 0.1
  10. 10. Cos(x) Sin(x) 0.9 Sin(x) 0.1 The angle is very small This means I can use the approximation... Sin(x) x 0.1 ... which gives the answer in radians.
  11. 11. Cos(x) Sin(x) 0.9 Sin(x) 0.1 The angle is very small This means I can use the approximation... Sin(x) x 0.1 x 0.1 rad
  12. 12. Cos(x) Sin(x) 0.9 Sin(x) 0.1 To get the answer in degrees, simply multiply by 57. x 5.7O 0.1 x 0.1 rad
  13. 13. Cos(x) Sin(x) 0.9 Sin(x) 0.1 There will be some error, because the Cosine line curves down from 1 just a little bit. But the error is slight. x 5.7O 0.1Check it out on your calculator. The error is just 0.3 degrees.
  14. 14. Cos(x) Sin(x) 0.9 Sin(x) 0.1So some questions now:Is this a quick and dirty approximation or is it a legitimate time-saver?How important is it to get that extra 0.3 degrees of accuracy?Should we be teaching methods like thisalongside the ‘proper’ methods?
  15. 15. Cos(x) Sin(x) 0.9Cos(x) Sin(x) 2Cos ( x 45 O )Here’s what the proper method looks like:Transform the Cos-Sin term into a single term.
  16. 16. Cos(x) Sin(x) 0.9Cos(x) Sin(x) 2Cos ( x 45 O ) 0.9 0.9 Cos ( x 45 O ) 0.6364 2 x 45O 50.48O x 5.48O
  17. 17. The method works brilliantly because of that clever transform.But I can’t always remember that transform so I’d have to look it upor work it out. After that, I’d need a calculator to invert the Cosine.
  18. 18. The graphical approach doesn’t always lead to an easy answer,but when it does work, it’s very fast and very easy. I like it.
  19. 19. [END]

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