Simplify arcsec((2 square root of 3)/3)

Problem

arcsec((2√(,3))/3)

Solution

1. Rewrite the expression using the definition of the inverse secant function. We are looking for an angle θ such that sec(θ)=(2√(,3))/3 within the range [0,π/2)∪(π/2,π]

sec(θ)=(2√(,3))/3

2. Convert the secant function to the cosine function using the reciprocal identity cos(θ)=1/sec(θ)

cos(θ)=3/(2√(,3))

3. Rationalize the denominator of the fraction to identify a standard value on the unit circle.

cos(θ)=(3√(,3))/(2⋅3)

cos(θ)=√(,3)/2

4. Identify the angle θ in the interval [0,π] that satisfies the cosine equation.

θ=π/6

Final Answer

arcsec((2√(,3))/3)=π/6

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