Find the Exact Value sec(315 degrees )

Problem

sec(315^∘)

Solution

1. Identify the reference angle by finding the difference between 315^∘ and the nearest x-axis angle, which is 360^∘

(θ_ref)=360^∘−315^∘=45^∘

2. Determine the quadrant of the angle. Since 315^∘ is between 270^∘ and 360^∘ it lies in Quadrant IV.

3. Determine the sign of the secant function in Quadrant IV. Since sec(θ)=1/cos(θ) and cosine is positive in Quadrant IV, secant is also positive.

sec(315^∘)=sec(45^∘)

4. Apply the reciprocal identity for secant.

sec(45^∘)=1/cos(45^∘)

5. Substitute the known value cos(45^∘)=√(,2)/2

sec(315^∘)=1/√(,2)/2

6. Simplify the expression by multiplying by the reciprocal.

sec(315^∘)=2/√(,2)

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

sec(315^∘)=(2√(,2))/2=√(,2)

Final Answer

sec(315^∘)=√(,2)

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