Simplify sec(150)

Problem

sec(150^∘)

Solution

1. Identify the reference angle for 150^∘ Since 150^∘ is in the second quadrant, the reference angle is 180^∘−150^∘=30^∘

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

3. Apply the reference angle and sign to the expression.

sec(150^∘)=−sec(30^∘)

4. Evaluate the secant of the reference angle. Since cos(30^∘)=√(,3)/2 the secant is the reciprocal.

sec(30^∘)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

6. Combine the value with the negative sign determined in step 2.

sec(150^∘)=−(2√(,3))/3

Final Answer

sec(150^∘)=−(2√(,3))/3

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