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

Problem

sec(−(9*π)/4)

Solution

1. Apply the even-odd property of the secant function, which states sec(−θ)=sec(θ)

sec(−(9*π)/4)=sec((9*π)/4)

2. Find the coterminal angle within the interval [0,2*π) by subtracting 2*π from the argument.

(9*π)/4−2*π=(9*π)/4−(8*π)/4

(9*π)/4−(8*π)/4=π/4

3. Identify the reciprocal identity for secant to express the value in terms of cosine.

sec(π/4)=1/cos(π/4)

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

sec(π/4)=1/√(,2)/2

5. Simplify the fraction by multiplying by the reciprocal.

1/√(,2)/2=2/√(,2)

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

(2√(,2))/2=√(,2)

Final Answer

sec(−(9*π)/4)=√(,2)

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