Find the Exact Value sec(240)

Problem

sec(240^∘)

Solution

1. Identify the reference angle by determining the quadrant in which 240^∘ lies. Since 180^∘<240^∘<270^∘ the angle is in Quadrant III.

2. Calculate the reference angle θ^′ using the formula for Quadrant III: θ^′=θ−180^∘

θ^′=240^∘−180^∘=60^∘

3. Determine the sign of the secant function in Quadrant III. Since sec(θ)=1/cos(θ) and cosine is negative in Quadrant III, the result will be negative.

sec(240^∘)=−sec(60^∘)

4. Evaluate the exact value using the reciprocal identity sec(60^∘)=1/cos(60^∘)

cos(60^∘)=1/2

sec(60^∘)=1/(1/2)=2

5. Apply the negative sign determined in step 3 to find the final value.

sec(240^∘)=−2

Final Answer

sec(240^∘)=−2

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