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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