Find the Exact Value sec(-(3pi)/4)

Problem

sec(−(3*π)/4)

Solution

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

sec(−(3*π)/4)=sec((3*π)/4)

2. Identify the reference angle for (3*π)/4 in the second quadrant.

(θ_ref)=π−(3*π)/4

(θ_ref)=π/4

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

sec((3*π)/4)=−sec(π/4)

4. Evaluate the secant of the reference angle using the reciprocal identity sec(θ)=1/cos(θ)

cos(π/4)=√(,2)/2

sec(π/4)=2/√(,2)

5. Rationalize the denominator to simplify the expression.

2/√(,2)=√(,2)

6. Combine the results by applying the negative sign determined in step 3.

sec((3*π)/4)=−√(,2)

Final Answer

sec(−(3*π)/4)=−√(,2)

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