Find the Exact Value sec((9pi)/4)

Problem

sec((9*π)/4)

Solution

1. Find the coterminal angle by subtracting 2*π from the given angle to find an equivalent angle within the interval [0,2*π)

(9*π)/4−2*π=(9*π)/4−(8*π)/4

(9*π)/4−(8*π)/4=π/4

2. Apply the reciprocal identity for the secant function, which states that sec(θ)=1/cos(θ)

sec((9*π)/4)=sec(π/4)

sec(π/4)=1/cos(π/4)

3. Substitute the known value for the cosine of π/4 which is √(,2)/2

cos(π/4)=√(,2)/2

sec(π/4)=1/√(,2)/2

4. Simplify the expression by taking the reciprocal of the fraction.

1/√(,2)/2=2/√(,2)

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

2/√(,2)⋅√(,2)/√(,2)=(2√(,2))/2

(2√(,2))/2=√(,2)

Final Answer

sec((9*π)/4)=√(,2)

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