1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Three forms

  1. Jan 8, 2008 #1
    How can there be three forms of the double-angle formula for cos 20?
  2. jcsd
  3. Jan 8, 2008 #2

    Gib Z

    User Avatar
    Homework Helper

    Just as (1+1) = 2, they are the same thing, just written in a different form.

    Do you know the 3 forms? If you do, post them here, and we can work out how they are the same.
  4. Jan 8, 2008 #3
  5. Jan 8, 2008 #4

    Gib Z

    User Avatar
    Homework Helper

    Well we have [tex]\cos 2a = \cos^2 a - \sin^2 a[/tex].

    We also know from another well known identity that [tex]\sin^2 a + \cos^2 a =1 [/tex] for all values of a. We can see that because it we have a right angled triangle and label one other angle as a, the adjacent side as A, the opposite side as O and the hypotenuse as H, by Pythagoras [itex]O^2 + A^2 = H^2[/itex].

    If we divide everything by H^2, [tex]\left(\frac{O}{H}\right)^2 +\left(\frac{A}{H}\right)^2 = 1[/tex].

    But since O/H is sin a, and A/H is cos a, we have that nice relationship.

    Now, since [itex]\sin^2 a + \cos^2 a =1[/tex], we can take either sin^2 a or cos^2 a to the other side: [tex]\cos^2 a = 1 - \sin^2 a[/tex] and [tex]\sin^2 a = 1 - \cos^2 a[/tex].

    Now from the original [tex]\cos^2 a - \sin^2 a[/tex], we replace sin^2 a with (1 - cos^2 a), we get the second form, and if we replace cos^2 a with (1 - sin^2 a) we get the third form =]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook