Find the Exact Value sec(135)

Problem

sec(135^∘)

Solution

1. Identify the reference angle for 135^∘ by subtracting it from 180^∘ because it is in the second quadrant.

(θ_ref)=180^∘−135^∘=45^∘

2. 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^∘)

3. Recall the exact value of cos(45^∘) which is √(,2)/2

cos(45^∘)=√(,2)/2

4. Calculate the reciprocal to find sec(45^∘)

sec(45^∘)=1/cos(45^∘)=2/√(,2)=√(,2)

5. Apply the negative sign determined in step 2 to find the final value.

sec(135^∘)=−√(,2)

Final Answer

sec(135^∘)=−√(,2)

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