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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