Find the Exact Value sec((19pi)/4)

Problem

sec((19*π)/4)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (8*π)/4 until the angle is within the interval [0,2*π)

(19*π)/4−2*π=(19*π)/4−(8*π)/4=(11*π)/4

(11*π)/4−2*π=(11*π)/4−(8*π)/4=(3*π)/4

2. Identify the reciprocal identity for the secant function.

sec((3*π)/4)=1/cos((3*π)/4)

3. Determine the cosine value for the angle (3*π)/4 which is in the second quadrant where cosine is negative.

cos((3*π)/4)=−√(,2)/2

4. Calculate the reciprocal to find the final value.

sec((3*π)/4)=1/(−√(,2)/2)

sec((3*π)/4)=−2/√(,2)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

−(2√(,2))/2=−√(,2)

Final Answer

sec((19*π)/4)=−√(,2)

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