Determine the exact value of tan(38pi/12) or tan(19pi/6)

19π/6 = 3π + π/6

The tan function has period π, so
tan(19π/6) = tan(π/6) = 1/√3

To determine the exact value of tan(38π/12) or tan(19π/6), we can use the following formula:

tan(x) = sin(x) / cos(x)

First, let's find the values of sin(38π/12) and cos(38π/12):

sin(38π/12) = sin(19π/6) = sin(π/6) = 1/2
cos(38π/12) = cos(19π/6) = cos(π/6) = √3/2

Now we can substitute these values back into the original formula:

tan(38π/12) = sin(38π/12) / cos(38π/12) = (1/2) / (√3/2)

To simplify this further, we can multiply the numerator and denominator by the reciprocal of √3/2, which is 2/√3:

tan(38π/12) = (1/2) / (√3/2) * (2/√3)

Simplifying the expression:

tan(38π/12) = (1/2) * (2/√3) / (√3/2)

The 2's cancel out:

tan(38π/12) = 1 / √3

To rationalize the denominator, we multiply both the numerator and denominator by √3:

tan(38π/12) = (1 / √3) * (√3 / √3) = √3 / 3

Therefore, the exact value of tan(38π/12) or tan(19π/6) is √3/3.

To determine the exact value of tan(38π/12) or tan(19π/6), we need to simplify the angle to an equivalent angle within one full circle of 2π radians.

First, let's simplify the angle 38π/12:

38π/12 = 19π/6

Now, we can find the reference angle by subtracting the nearest multiple of π/2 from 19π/6:

19π/6 - 3π/2 = π/6

Since the reference angle is π/6, we can use the trigonometric identity for tangent:

tan(π/6) = √3/3

Therefore, the exact value of tan(38π/12) or tan(19π/6) is √3/3.