Find the Exact Value sec(-315 degrees )

Problem

sec(−315)

Solution

1. Use the even-odd property of the secant function, which states sec(−θ)=sec(θ)

sec(−315)=sec(315)

2. Find the coterminal angle or identify the position on the unit circle. The angle 315 is in the fourth quadrant.

315=360−45

3. Determine the reference angle for 315 which is 45

(θ_ref)=360−315=45

4. Apply the secant definition in the fourth quadrant. Since sec(θ)=1/cos(θ) and cosine is positive in the fourth quadrant, the value remains positive.

sec(315)=sec(45)

5. Evaluate the exact value using the known value for cos(45)=√(,2)/2

sec(45)=1/cos(45)

sec(45)=2/√(,2)

6. Rationalize the denominator to simplify the expression.

2/√(,2)⋅√(,2)/√(,2)=(2√(,2))/2

(2√(,2))/2=√(,2)

Final Answer

sec(−315)=√(,2)

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