Find the Exact Value sec((17pi)/6)

Problem

sec((17*π)/6)

Solution

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

(17*π)/6−2*π=(17*π)/6−(12*π)/6

(17*π)/6−(12*π)/6=(5*π)/6

2. Identify the reference angle for (5*π)/6 which is located in the second quadrant.

π−(5*π)/6=π/6

3. Determine the sign of the secant function in the second quadrant. Since cos(θ) is negative in the second quadrant, sec(θ) is also negative.

sec((5*π)/6)=−sec(π/6)

4. Evaluate the secant using the reciprocal of the cosine of the reference angle.

cos(π/6)=√(,3)/2

sec(π/6)=2/√(,3)

5. Rationalize the denominator to find the exact value.

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

6. Apply the negative sign determined in step 3.

sec((17*π)/6)=−(2√(,3))/3

Final Answer

sec((17*π)/6)=−(2√(,3))/3

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