Unravelling the Mystery of the 3 Forms of Double-Angle Formula for Cos 20

[danglade]

How can there be three forms of the double-angle formula for cos 20?

 

[Gib Z]

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.

 

[danglade]

http://www.themathpage.com/aTrig/Trig_IMG/125.gif

 

[Gib Z]

Well we have $$\cos 2a = \cos^2 a - \sin^2 a$$.

We also know from another well known identity that $$\sin^2 a + \cos^2 a =1$$ 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 $O^2 + A^2 = H^2$.

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

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: $$\cos^2 a = 1 - \sin^2 a$$ and $$\sin^2 a = 1 - \cos^2 a$$.

Now from the original $$\cos^2 a - \sin^2 a$$, 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 =]