Find the Exact Value sec(30)

Problem

sec(30^∘)

Solution

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

sec(θ)=1/cos(θ)

2. Substitute the given angle of 30^∘ into the identity.

sec(30^∘)=1/cos(30^∘)

3. Determine the exact value of cos(30^∘) using the unit circle or a special right triangle (30^∘−60^∘−90^∘.

cos(30^∘)=√(,3)/2

4. Calculate the reciprocal by dividing 1 by the value of the cosine.

sec(30^∘)=1/√(,3)/2

5. Simplify the fraction by multiplying by the reciprocal of the denominator.

sec(30^∘)=2/√(,3)

6. Rationalize the denominator by multiplying the numerator and the denominator by √(,3)

sec(30^∘)=(2√(,3))/3

Final Answer

sec(30^∘)=(2√(,3))/3

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