Question 207450
The only "exact values" for Trigonometric functions which can be found are for 0, 30, 45, 60, 90 degress (and for angles whose reference angles are these). So the only way to find the excat value of tan(15) is to express the tan(15) in terms of these other angles.

Since 15 = 1/2 of 30 we could use the 1/2 angle formula:
{{{tan(1/2)x = (1 - cos(x))/(sin(x))}}}
where x = 30:
{{{tan(1/2)30 = (1 - cos(30))/(sin(30))}}}
Since {{{cos(30) = sqrt(3)/2}}} and {{{sin(30) = 1/2}}}:
{{{tan(15) = ((1 - sqrt(3)/2))/(1/2)}}}
Change the division by 1/2 to mutiplying by 2/1:
{{{tan(15) = (1 - sqrt(3)) * (2/1)}}}
{{{tan(15) = 2 - sqrt(3)}}}<br>

Another way would be to use the fact that 15 = 45 - 30 and use the difference formula
{{{tan(a-b) = (tan(a) - tan(b))/(1+tan(a) tan(b))}}}
with a = 45 and b = 30:
{{{tan(45-30) = (tan(45) - tan(30))/(1 + tan(45) tan(30))}}}
Since {{{tan(45) = 1}}} and {{{tan(30) = sqrt(3)/3}}}:
{{{tan(15) = (1 - sqrt(3)/3)/(1 + 1 * (sqrt(3)/3))}}}
Multiplying the top and bottom by 3:
{{{tan(15) = (3 - sqrt(3))/(3 + sqrt(3))}}}
Next we rationalize the denominator by mulitplying by its conjugate: {{{(3 - sqrt(3))}}}:
{{{tan(15) = (9 - 6*sqrt(3) + 3)/(9 - 3)}}}
{{{tan(15) = (12 - 6*sqrt(3))/6}}}
Factoring out 6 on top:
{{{tan(15) = (6(2 - sqrt(3)))/6}}}
Canceling the common factor of 6:
{{{tan(15) = 2 - sqrt(3)}}}