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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