Find the Exact Value sec(390)

Problem

sec(390^∘)

Solution

1. Find the coterminal angle by subtracting 360^∘ from the given angle to find an equivalent angle within the first rotation.

390^∘−360^∘=30^∘

2. Apply the reciprocal identity for the secant function, which states that sec(θ)=1/cos(θ)

sec(390^∘)=sec(30^∘)

sec(30^∘)=1/cos(30^∘)

3. Substitute the known value for cos(30^∘) which is √(,3)/2

sec(30^∘)=1/√(,3)/2

4. Simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

sec(30^∘)=2/√(,3)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

sec(390^∘)=(2√(,3))/3

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