Find the Exact Value sec((2pi)/3)

Problem

sec((2*π)/3)

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 reference angle for (2*π)/3 in the unit circle.

(θ_ref)=π−(2*π)/3=π/3

3. Evaluate the cosine of the angle by considering its quadrant. Since (2*π)/3 is in Quadrant II, the cosine value is negative.

cos((2*π)/3)=−cos(π/3)

4. Substitute the known value of cos(π/3)=1/2 into the expression.

cos((2*π)/3)=−1/2

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

sec((2*π)/3)=1/(−1/2)

6. Simplify the fraction to find the final exact value.

sec((2*π)/3)=−2

Final Answer

sec((2*π)/3)=−2

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