Find the Exact Value sec(135 degrees )

Problem

sec(135^∘)

Solution

1. Identify the relationship between the secant function and the cosine function using the reciprocal identity.

sec(135^∘)=1/cos(135^∘)

2. Determine the reference angle for 135^∘ which is located in the second quadrant.

(θ_ref)=180^∘−135^∘=45^∘

3. Evaluate the cosine of 135^∘ by considering the sign in the second quadrant, where cosine is negative.

cos(135^∘)=−cos(45^∘)

cos(135^∘)=−√(,2)/2

4. Substitute the value of cos(135^∘) back into the reciprocal identity.

sec(135^∘)=1/(−√(,2)/2)

5. Simplify the expression by multiplying by the reciprocal.

sec(135^∘)=−2/√(,2)

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

sec(135^∘)=−(2√(,2))/2

sec(135^∘)=−√(,2)

Final Answer

sec(135^∘)=−√(,2)

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