Find the Exact Value sec(120 degrees )

Problem

sec(120^∘)

Solution

1. Identify the reference angle for 120^∘ by subtracting it from 180^∘ because it is in the second quadrant.

180^∘−120^∘=60^∘

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

sec(120^∘)=−sec(60^∘)

3. Recall the reciprocal identity for secant.

sec(60^∘)=1/cos(60^∘)

4. Substitute the known exact value for cos(60^∘)=1/2

sec(60^∘)=1/(1/2)=2

5. Apply the negative sign determined in step 2 to find the final value.

sec(120^∘)=−2

Final Answer

sec(120^∘)=−2

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