Find the Exact Value sec(-510)

Problem

sec(−510^∘)

Solution

1. Find the coterminal angle by adding 360^∘ repeatedly until the angle is within the range [0^∘,360^∘)

−510^∘+360^∘=−150^∘

−150^∘+360^∘=210^∘

2. Identify the quadrant for 210^∘ Since 180^∘<210^∘<270^∘ the angle is in Quadrant III.

3. Determine the reference angle for 210^∘ in Quadrant III.

(θ_ref)=210^∘−180^∘=30^∘

4. Apply the secant definition and consider the sign in Quadrant III. In Quadrant III, the secant function is negative.

sec(210^∘)=−sec(30^∘)

5. Evaluate the exact value using the reciprocal identity sec(θ)=1/cos(θ)

cos(30^∘)=√(,3)/2

sec(30^∘)=2/√(,3)

6. Rationalize the denominator to find the final value.

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

sec(−510^∘)=−(2√(,3))/3

Final Answer

sec(−510^∘)=−(2√(,3))/3

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