Question

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

Answer 1

`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`