Find the Exact Value sec(120)

Problem

sec(120^∘)

Solution

1. Identify the reference angle by finding the distance from the angle to the nearest x-axis. Since 120^∘ is in the second quadrant, the reference angle is 180^∘−120^∘=60^∘

2. Determine the sign of the secant function in the second quadrant. In the second quadrant, the cosine function is negative, and since sec(θ)=1/cos(θ) the secant function is also negative.

3. Recall the exact value of cos(60^∘)

cos(60^∘)=1/2

4. Apply the reciprocal identity to find the secant of the reference angle.

sec(60^∘)=1/cos(60^∘)=2

5. Combine the value with the appropriate sign for the second quadrant.

sec(120^∘)=−2

Final Answer

sec(120^∘)=−2

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