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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