Simplify tan(-150)

Problem

tan(−150^∘)

Solution

1. Apply the odd function property of the tangent function, which states that tan(−θ)=−tan(θ)

tan(−150^∘)=−tan(150^∘)

2. Find the reference angle for 150^∘ by subtracting it from 180^∘ since it is in the second quadrant.

Reference Angle=180^∘−150^∘=30^∘

3. Determine the sign of the tangent function in the second quadrant. Since tangent is negative in the second quadrant, tan(150^∘)=−tan(30^∘)

−tan(150^∘)=−(−tan(30^∘))

4. Simplify the signs to show that the expression is equivalent to the tangent of the reference angle.

−(−tan(30^∘))=tan(30^∘)

5. Evaluate the exact value of tan(30^∘) using the unit circle or special triangles.

tan(30^∘)=1/√(,3)

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

1/√(,3)=√(,3)/3

Final Answer

tan(−150^∘)=√(,3)/3

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