Find the Exact Value sec(405)

Problem

sec(405^∘)

Solution

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

405^∘−360^∘=45^∘

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

sec(405^∘)=sec(45^∘)

sec(45^∘)=1/cos(45^∘)

3. Substitute the known value for cos(45^∘) from the unit circle.

cos(45^∘)=√(,2)/2

4. Simplify the expression by taking the reciprocal of the fraction.

1/√(,2)/2=2/√(,2)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

2/√(,2)⋅√(,2)/√(,2)=(2√(,2))/2

(2√(,2))/2=√(,2)

Final Answer

sec(405^∘)=√(,2)

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