Find the Exact Value sec(330 degrees )

Problem

sec(330^∘)

Solution

1. Identify the reference angle by finding the difference between 330^∘ and the nearest x-axis value, which is 360^∘

360^∘−330^∘=30^∘

2. Determine the quadrant of the angle. Since 330^∘ is between 270^∘ and 360^∘ it lies in Quadrant IV.

3. Apply the sign rule for the secant function. In Quadrant IV, the cosine function is positive, and since sec(θ)=1/cos(θ) the secant function is also positive.

sec(330^∘)=sec(30^∘)

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

cos(30^∘)=√(,3)/2

5. Calculate the reciprocal to find the exact value.

sec(30^∘)=2/√(,3)

6. 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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