Find the Exact Value sec(315)

Problem

sec(315)

Solution

1. Identify the reference angle by finding the difference between 315 and 360 since the angle is in the fourth quadrant.

(θ_ref)=360−315=45

2. Determine the sign of the secant function in the fourth quadrant. Since sec(θ)=1/cos(θ) and cosine is positive in the fourth quadrant, the result will be positive.

sec(315)=sec(45)

3. Apply the reciprocal identity to relate secant to cosine.

sec(45)=1/cos(45)

4. Substitute the known exact value for cos(45)=√(,2)/2

sec(315)=1/√(,2)/2

5. Simplify the fraction by multiplying by the reciprocal.

sec(315)=2/√(,2)

6. 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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