Find the Exact Value sec((29pi)/6)

Problem

sec((29*π)/6)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (12*π)/6 from the given angle until it is within the range [0,2*π)

(29*π)/6−2*((12*π)/6)=(29*π)/6−(24*π)/6

(29*π)/6≡(5*π)/6

2. Identify the reciprocal relationship for the secant function.

sec((5*π)/6)=1/cos((5*π)/6)

3. Determine the cosine value for the angle (5*π)/6 which is in the second quadrant where cosine is negative.

cos((5*π)/6)=−√(,3)/2

4. Calculate the reciprocal to find the final value.

sec((5*π)/6)=1/(−√(,3)/2)

sec((5*π)/6)=−2/√(,3)

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

−2/√(,3)⋅√(,3)/√(,3)=−(2√(,3))/3

Final Answer

sec((29*π)/6)=−(2√(,3))/3

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