Question 515214
The half angle formula for tangent can be written in two different but equally simple ways:

{{{tan (theta/2) = (1 - cos (theta))/(sin (theta)) = (sin (theta))/(1 + cos (theta)) }}}

We'll use the first formula, but you could just as easily use the second and get the same answer.  We want to find the tangent of {{{(3/8) pi}}}.  The angle {{{(3/8) pi}}} is one half of the angle {{{(3/4) pi}}}, since:

{{{ ((3/4) pi)/ 2 = (1/2) * (3/4) pi = (3/8) pi}}}

So we'll try applying the half angle formula for tangent using {{{theta = (3/4) pi}}}.

{{{ tan ((3/8) pi) = tan (((3/4) pi)/ 2) = (1 - cos ((3/4) pi))/ (sin ((3/4) pi)) }}}

Using the unit circle, we can see that {{{cos ((3/4) pi) = - (sqrt(2))/2 }}} and {{{sin ((3/4) pi) = (sqrt(2))/2}}} (this makes sense since the angle {{{(3/4) pi}}} is in the second quadrant and makes a 45 degree angle with the negative x-axis).  So we substitute in:

{{{(1 - cos ((3/4) pi))/ (sin ((3/4) pi)) = (1 - (-(sqrt(2))/2))/(sqrt(2)/2)}}}

Simplifying, we get:

{{{(1 + (sqrt(2))/2)/((sqrt(2))/2) }}} (canceling the negatives in the numerator)

{{{ (2/(sqrt(2)) * (1 + (sqrt(2))/2)) }}} (getting rid of the complex fraction by multiplying by the reciprocal of the denominator instead)

{{{ 2/(sqrt(2)) + 1 }}} (distributing the {{{2/(sqrt(2))}}})

{{{ sqrt(2) + 1 }}} (simplifying {{{2/(sqrt(2))}}} to be {{{sqrt(2)}}})

So the tangent of {{{(3/8) pi}}} is {{{1 + sqrt(2)}}} exactly.  This should match what you get on your calculator, and it also makes sense: {{{(3/8) pi}}} is in the first quadrant, so its tangent should be positive, and it is larger angle than 45 degrees, so its tangent should be larger than 1.