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We wish to find the value of the expression `cos((23pi)/12)`.
The domain of `x` over one whole period of the cosine function `cos(x)` is ` `0 to `2pi` . If we look at a graph of a single period of the cosine function we see that it is symmetric about `x=pi`:
so that `cos(x) = cos(2pi-x)`.
Now, `(23pi)/12 = ((24-1)pi)/12 = 2pi - pi/12`
Therefore, if `x= pi/12`,
`cos(pi/12) = cos(2pi-pi/12) = cos((23pi)/12)`
Reading off the graph, `cos(pi/12)` is very close to `1`. If you put it into your calculator (as 'cos' 'bracket' 'pi' / 12 'close bracket' ) you will get `cos(pi/12) = 0.966` to 3 significant figures.
Answer ` ``cos((23pi)/12) = cos(pi/12) = 0.966` to 3 significant figures
You may write the argument of cosine, such that:
`23pi/12 = 24pi/12 - pi/12`
Taking the cosine yields:
`cos(23pi/12) = cos(2pi - pi/12) => cos(23pi/12) = cos (pi/12)`
Using the half angle identity, yields:
`cos (pi/12) = sqrt((1 + cos(pi/6))/2)`
`cos (pi/12) = sqrt((1 + sqrt3/2)/2) => cos (pi/12) = sqrt(2 + sqrt3)/2`
Hence, evaluating the given cosine, using the half angle formula, yields `cos(23pi/12) = sqrt(2 + sqrt3)/2.`
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