Evaluate tan(390 degrees )

Problem

tan(390^∘)

Solution

1. Identify the periodicity of the tangent function. Since the tangent function has a period of 180^∘ we can also use the fact that any trigonometric function repeats every 360^∘ (one full revolution).

2. Subtract the full revolution from the angle to find a coterminal angle between 0^∘ and 360^∘

θ=390^∘−360^∘

θ=30^∘

3. Apply the trigonometric identity for coterminal angles.

tan(390^∘)=tan(30^∘)

4. Evaluate the tangent of the reference angle using known values from the unit circle.

tan(30^∘)=sin(30^∘)/cos(30^∘)

tan(30^∘)=(1/2)/(√(,3)/2)

tan(30^∘)=1/√(,3)

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

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

tan(390^∘)=√(,3)/3

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