Find the Value Using the Unit Circle sec(120 degrees )

Problem

sec(120^∘)

Solution

1. Identify the relationship between the secant function and the cosine function using the reciprocal identity.

sec(θ)=1/cos(θ)

2. Determine the coordinates of the angle 120^∘ on the unit circle. The angle 120^∘ is in the second quadrant, where the xcoordinate represents the cosine value.

(x,y)=(cos(120^∘),sin(120^∘))

3. Find the cosine value using the reference angle. The reference angle for 120^∘ is 180^∘−120^∘=60^∘ In the second quadrant, cosine is negative.

cos(120^∘)=−cos(60^∘)

4. Substitute the known value of cos(60^∘)=1/2 into the expression.

cos(120^∘)=−1/2

5. Calculate the secant value by taking the reciprocal of the cosine value.

sec(120^∘)=1/(−1/2)

6. Simplify the fraction to find the final result.

sec(120^∘)=−2

Final Answer

sec(120^∘)=−2

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