Simplify sec((5pi)/6)

Problem

sec((5*π)/6)

Solution

1. Identify the definition of the secant function in terms of the cosine function.

sec(θ)=1/cos(θ)

2. Determine the reference angle for (5*π)/6 by subtracting it from π

π−(5*π)/6=π/6

3. Evaluate the cosine of the reference angle π/6

cos(π/6)=√(,3)/2

4. Determine the sign of the cosine function in the second quadrant, where (5*π)/6 is located.

cos((5*π)/6)=−√(,3)/2

5. Substitute the value of the cosine into the secant formula.

sec((5*π)/6)=1/(−√(,3)/2)

6. Simplify the fraction by multiplying by the reciprocal.

sec((5*π)/6)=−2/√(,3)

7. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

sec((5*π)/6)=−(2√(,3))/3

Final Answer

sec((5*π)/6)=−(2√(,3))/3

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