Cosine Approximations

Date: 12/29/97 at 16:11:56
From: phil
Subject: Cosine of 40 degrees

You derived a cubic equation for cosine of 40 degrees. Did you know 
the following approximation?

cos (2x/3) = approx 0.5*(1 + cos x)
so if x = 60 degrees, cos 40 = approx 0.5*(1+0.5) = 0.75 versus
actual cos 40 = 0.76604, for 2 percent accuracy.

Phil F.

Date: 01/03/98 at 04:19:19
From: Doctor Pete
Subject: Re: Cosine of 40 degrees

Hi,

This approximation can be derived as well:  Let k = x/3.  Then

     Cos[3k] = Cos[k + 2k]
             = Cos[k]Cos[2k] - Sin[k]Sin[2k]
             = Sqrt[(1+Cos[2k])/2]Cos[2k]
                  - Sqrt[(1-Cos[2k])/2]Sqrt[1-Cos[2k]^2]
             = Sqrt[(1+m)/2]m - Sqrt[(1-m)/2]Sqrt[1-m^2]

where m = Cos[2k].  Simplifying, we obtain

     Cos[3k] = (Sqrt[1+m]m - Sqrt[1+m](1-m))/Sqrt[2]
             = Sqrt[1+m](m - (1-m))/Sqrt[2]
             = Cos[k](2*Cos[2k]-1).

Now, suppose that k is a small positive quantity.  Then

     Cos[k] ~ 1,

and so Cos[3k] ~ 2*Cos[2k]-1, or

     (1+Cos[x])/2 ~ Cos[2x/3].

In particular, when x = Pi/3 = 60 degrees, then we obtain the 
approximation you mentioned.

-Doctor Pete,  The Math Forum
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