Find the Exact Value sec(675)

Problem

sec(675^∘)

Solution

1. Find the coterminal angle by subtracting 360^∘ from the given angle until it is within the range [0^∘,360^∘)

675^∘−360^∘=315^∘

2. Identify the quadrant for 315^∘ Since 270^∘<315^∘<360^∘ the angle lies in Quadrant IV.

3. Determine the reference angle for an angle in Quadrant IV using the formula (θ_ref)=360^∘−θ

360^∘−315^∘=45^∘

4. Apply the secant identity sec(θ)=1/cos(θ) In Quadrant IV, the cosine (and therefore the secant) is positive.

sec(675^∘)=sec(45^∘)

5. Evaluate the exact value using the known trigonometric values for 45^∘

cos(45^∘)=√(,2)/2

sec(45^∘)=2/√(,2)

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

2/√(,2)⋅√(,2)/√(,2)=(2√(,2))/2=√(,2)

Final Answer

sec(675^∘)=√(,2)

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