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Trigonometry (Double angle formula)

  1. Apr 11, 2013 #1

    trollcast

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    1. The problem statement, all variables and given/known data
    Calculate the value of [itex]\tan(\frac{\pi}{12})[/itex] without the use of a calculator and showing all steps.

    2. Relevant equations

    Compound angle formula / Double angle formula

    3. The attempt at a solution

    Using an equilateral triangle of side length 2 I've shown that:
    $$\tan(\frac{\pi}{6})=\frac{1}{\sqrt{3}}$$
    Then using the double angle formula:
    $$\tan(2\theta)=\frac{2\tan(\theta)}{1-\tan^2(\theta)}$$
    Substituting the value for [itex]\tan(\frac{\pi}{12})[/itex] in as 2θ and solving for [itex]\tan(\theta)[/itex]:
    $$\tan(\frac{\pi}{12})=\pm2-\sqrt(3)$$

    However how do I show that I only need to take the positive solution?

    Thanks
     
  2. jcsd
  3. Apr 11, 2013 #2

    micromass

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    You know that ##\pi/12## is in the first quadrant. What must the tangent be of an angle in the first quadrant? Positive or negative?
     
  4. Apr 11, 2013 #3

    trollcast

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    Thanks micro don't know how I missed that.

    So the tangent should be positive since its in the first quadrant, therefore the negative solution is not possible?
     
  5. Apr 11, 2013 #4

    micromass

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    That's it!
     
  6. Apr 11, 2013 #5

    trollcast

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    Thanks again, I just actually realised I could have done it by finding [itex]\tan(\frac{\pi}{4})[/itex] and then using the compound angle formula to calculate [itex]\tan(\frac{\pi}{4}-\frac{\pi}{6})[/itex] , which would only give one solution at the end.
     
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