Find the Exact Value sec(-330 degrees )

Problem

sec(−330)

Solution

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

sec(−330)=sec(330)

2. Find the coterminal angle or reference angle. Since 330 is in the fourth quadrant, the reference angle is 360−330=30

(θ_ref)=30

3. Determine the sign of the secant function in the fourth quadrant. Since cos(θ) is positive in the fourth quadrant, sec(θ) is also positive.

sec(330)=sec(30)

4. Evaluate the secant using the reciprocal identity sec(θ)=1/cos(θ)

sec(30)=1/cos(30)

5. Substitute the known value for cos(30)=√(,3)/2

sec(30)=1/√(,3)/2

6. Simplify the fraction by multiplying by the reciprocal.

sec(30)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

sec(−330)=(2√(,3))/3

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