Find the Exact Value sec(150)

Problem

sec(150^∘)

Solution

1. Identify the reference angle for 150^∘ by subtracting it from 180^∘

(θ_ref)=180^∘−150^∘=30^∘

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

sec(150^∘)=−sec(30^∘)

3. Recall the value of cos(30^∘) to find its reciprocal.

cos(30^∘)=√(,3)/2

4. Calculate the reciprocal to find sec(30^∘)

sec(30^∘)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

6. Apply the negative sign determined in step 2.

sec(150^∘)=−(2√(,3))/3

Final Answer

sec(150^∘)=−(2√(,3))/3

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