find the exact circular function value for each of the following sec 23pi/6

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember the definition of secant function, such that:

`sec alpha = 1/(cos alpha)`

Hence, you need to determine `sec (23pi/6)` such that:

`sec (23pi/6) = 1/(cos (23pi/6))`

You need to evaluate `23pi/6` such that:

`23pi/6 = 24pi/6 - pi/6 => 23pi/6 = 4pi - pi/6`

`(cos (23pi/6)) = cos (4pi - pi/6)`

You need to expand `cos (4pi - pi/6)` using the following formula, such that:

`cos(alpha - beta) = cos alpha*cos beta + sin alpha*sin beta`

Reasoning by analogy, yields:

`cos (4pi - pi/6) = cos 4pi*cos (pi/6) + sin 4pi*sin pi/6`

`cos 4pi = cos 2*(2pi) = 2cos^2(2pi) - 1`

Since `cos 2pi = 1` yields:

`cos 4pi = 2 - 1 = 1`

Since `sin 4pi = 0` yields:

`cos (4pi - pi/6) = cos (pi/6) = sqrt3/2`

Hence, you may evaluate the secant of `(23pi/6)` , such that:

`sec (23pi/6) = 1/(sqrt3/2) => sec (23pi/6) = 2/sqrt3`

Using rationalization yields:

`sec (23pi/6) = 2sqrt3/3`

Hence, evaluating the given secant yields `sec (23pi/6) = 2sqrt3/3.`

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