Simplify tan(pi/6)

Problem

tan(π/6)

Solution

1. Identify the angle in degrees to better visualize the position on the unit circle.

π/6* radians=30

2. Recall the definition of the tangent function in terms of sine and cosine.

tan(θ)=sin(θ)/cos(θ)

3. Determine the exact values for the sine and cosine of the given angle from the unit circle.

sin(π/6)=1/2

cos(π/6)=√(,3)/2

4. Substitute these values into the tangent ratio.

tan(π/6)=1/2/√(,3)/2

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

tan(π/6)=1/2⋅2/√(,3)

tan(π/6)=1/√(,3)

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

tan(π/6)=(1⋅√(,3))/(√(,3)⋅√(,3))

tan(π/6)=√(,3)/3

Final Answer

tan(π/6)=√(,3)/3

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