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Why isn't cos(22.5°) = 2/√5?

  1. Dec 21, 2014 #1
    I've never really learned any trigonometry and I'm wondering if someone could tell me where I'm going wrong here.

    My reasoning is this:

    The hypotenuse of a 1,1,√2 right triangle is at a 45° angle to its base.

    Halving that angle should require that the base leg be doubled if the height leg is kept at 1, giving the new triangle lengths of 1,2,√5

    The cosine is adjacent/hypotenuse which would be 2/√5, yet calculators give a different answer. Why is this?
  2. jcsd
  3. Dec 21, 2014 #2


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    Your mistake is in thinking that a 1, 2, sqrt(5) triangle has one angle 22.5 degrees.
  4. Dec 21, 2014 #3
    So this assumption isn't correct?

    Edit: I see now that it's not but I don't understand why... Seems like it should be

    2nd edit: my problem was that I was equating angle with slope. I like math... Too bad I'm so bad at it! :/
    Last edited: Dec 21, 2014
  5. Dec 22, 2014 #4


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    The relation between angle and slope is

    $$\mathrm{slope}=\dfrac{\mathrm{rise}}{\mathrm{run}}=\dfrac{\Delta y}{\Delta x}=\dfrac{\sin(\theta)}{\cos(\theta)}=\tan(\theta)$$
  6. Dec 23, 2014 #5
    Draw a right angle triangle using protractor,scale,pen or pencil which I just did.By drawing the dimensions of right angle triangle of sides 1,2,## \sqrt 5 ## you will see that the angle made by hypotenuse and base is approximately 26.5° not your 22.5°.So, cos 26.5°= 2/##\sqrt 5## So the calculators were right and you may have made a minute mistake.
  7. Dec 23, 2014 #6
  8. Dec 23, 2014 #7


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