Find the Exact Value sec(-20)

Problem

sec(−210)

Solution

1. Apply the even-odd identity for the secant function, which states sec(−θ)=sec(θ)

sec(−210)=sec(210)

2. Find the reference angle for 210 Since the angle is in the third quadrant, subtract 180 from the angle.

Reference Angle=210−180=30

3. Determine the sign of the secant function in the third quadrant. Since cosine (and therefore secant) is negative in the third quadrant, the result will be negative.

sec(210)=−sec(30)

4. Evaluate the secant of the reference angle. Since cos(30)=√(,3)/2 the secant is the reciprocal.

sec(30)=2/√(,3)

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

2/√(,3)=(2√(,3))/3

6. Combine the value and the sign to find the final result.

sec(210)=−(2√(,3))/3

Final Answer

sec(−210)=−(2√(,3))/3

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