Find the Exact Value sec(-135)

Problem

sec(−135)

Solution

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

sec(−135)=sec(135)

2. Identify the reference angle for 135 Since 135 is in the second quadrant, the reference angle is 180−135=45

(θ_ref)=45

3. Determine the sign of the secant function in the second quadrant. Since sec(θ)=1/cos(θ) and cosine is negative in the second quadrant, secant is also negative.

sec(135)=−sec(45)

4. Evaluate the secant of the reference angle. We know cos(45)=√(,2)/2 so its reciprocal is 2/√(,2) which simplifies to √(,2)

sec(45)=√(,2)

5. Combine the value and the sign to find the final result.

sec(135)=−√(,2)

Final Answer

sec(−135)=−√(,2)

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