Question 625300
I assume that the instructions for the problem say to find the exact value. Otherwise we could just use our calculators and it wouldn't be much of a problem. (Please include the instructions so we don't have to guess.)<br>
Whenever the expression "exact value" is used in Trig. it is cods for "use special angles". 195 has a reference angle of 15 which is not a special angle. So 195 itself is not a special angle (which would have made this problem easy).<br>
Now we have to find a way to express 195 in terms of special angles. We have to ask ourselves:<ul><li>Is 195 the sum of two special angles?</li><li>Is 195 the difference of two special angles?</li><li>Is 195 the 1/2 of a special angle?</li><li>Is 195 the some other combination of special angles?</li></ul>If you're lucky, you will be able to answer "yes" to one of the first three questions. That is because we have formulas for sums, differences and halves of angles.<br>
With a little effort we should be able to answer "yes" to the first two equations:
195 = 150 + 45
and 
195 = 240 - 45
So we will be able to get an answer in two ways:
tan(195)
tan(150+45)
Using the formula {{{tan(A+B) = (tan(A)+tan(B))/(1-tab(A)*tan(B))}}} we get:
{{{(tan(150) + tan(45))/(1-tan(150)*tan(45))}}}
We should be able to find these tan's easily since they are all special angles:
{{{((-sqrt(3)/3) + (1))/(1-(-sqrt(3)/3)*(1))}}}
which simplifies as follows:
{{{(-sqrt(3)/3 + 1)/(1+sqrt(3)/3)}}}
Multiplying the numerator and denominator by 3:
{{{(-sqrt(3) + 3)/(3+sqrt(3))}}}
Rationalizing the denominator:
{{{((-sqrt(3) + 3)/(3+sqrt(3)))*((3-sqrt(3))/(3-sqrt(3)))}}}
{{{((-sqrt(3))*3 + (-sqrt(3))*(-sqrt(3)) + 3*3 + 3*(-sqrt(3))) /((3)^2-(sqrt(3))^2)}}}
{{{(-3sqrt(3) + 3 + 9 + (-3sqrt(3)))/(9-3)}}}
{{{(-6sqrt(3) + 12)/6}}}
{{{(6(-sqrt(3) + 2))/6}}}
{{{(cross(6)(-sqrt(3) + 2))/cross(6)}}}
{{{-sqrt(3) + 2}}}
<br>
Another solution using 240 - 45:
tan(195)
tan(240-45)
Using {{{tan(A-B) = (tan(A)-tan(B))/(1+tab(A)*tan(B))}}}
{{{(tan(240) - tan(45))/(1+tan(240)*tan(45))}}}
{{{((sqrt(3)) - (1))/(1+(sqrt(3))*(1))}}}
{{{(sqrt(3) - 1)/(1+sqrt(3))}}}
Rationalizing the denoinator:
{{{((sqrt(3) - 1)/(1+sqrt(3)))((1-sqrt(3))/(1-sqrt(3)))}}}
{{{(sqrt(3)*1 + sqrt(3)*(-sqrt(3))+(-1)*1+(-1)(-sqrt(3)))/((1)^2-(sqrt(3))^2)}}}
{{{(sqrt(3) + (-3)+(-1)+sqrt(3))/(1-3)}}}
{{{(2sqrt(3) + (-4))/(-2)}}}
{{{((-2)(-sqrt(3) + 2))/(-2)}}}
{{{(cross((-2))(-sqrt(3) + 2))/cross((-2))}}}
{{{-sqrt(3) + 2}}}<br>
So either way we find that the exact value of tan(195) is {{{-sqrt(3) + 2}}}.