How do I evaluate cos(pi/10) without using a calculator?

1 Answer
Mar 12, 2016

0.951 in 3 significants digits. (Details in Explanation)

Explanation:

For this purpose we can use the Maclaurin series or the Taylor series. Since pi/10~=0.314 is closer to 0 than to 1, its possible to use the Maclaurin series.

The Maclaurin series is given by
f(x)=f(0)+(f'(0))/(1!)*x+(f"''"(0))/(2!)*x^2+(f"'''"(0))/(3!)*x^3+...

Finding the coefficients
f(x)=cosx => f(x=0)=1
f'(x)=-sinx => f'(x=0)=0
f"''"(x)=-cosx => f"''"(x=0)=-1
f"'''"(x)=sinx => f"'''"(x=0)=0
f^(IV) (x)=cosx => f^(IV)(x=0)=1

And
f(x)=1-x^2/2+x^4/24+...

We can stop when the term of the series, different of zero, is inferior to the precision required.
Setting pi to 3 significant digits, it means that we consider pi=3.14 and pi/10=0.314

Estimating (with x=0.3) the function in 2 terms we have
f(x=0.3)=1-0.09/2=1-0.045=0.955
Estimating (with x=0.3) the function in 3 terms we have
f(x=0.3)=1-0.09/2+0.008/24=1-0.045+0.0003=0.9553
As we can see there's no need to work with this function with more than 2 terms since the desconsideration of the terms after the second one doesn't compromise the result with the intended accuracy.

So, with 3 significant digits, a good approximation of the function is
f(x)=1-x^2/2

Calculating x^2
> " "0,314
xx" "0,314
" "___ #" "1256 " + "314 " "942 " "_________________ " "0.098596#

Using x^2=0.0986 we get

f(x=0.314)=1-0.0986/2=1-0.0493=0.9507
In 3 significant digits:
f(x=0.314)=0.951