Expert: Sherman D. - 1/1/2007

Question
Sorry, but I have one last question. I asked this question to both experts Jack Cheng and Sherman D and you each gave me two very different answers that both seem right. Can you tell me which answer is right for sure and why. Thanks so much.


2)
Find the value of tan(105)
tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))

tan(105) = tan(45 + 60)

tan(45 + 60) = (tan(45) + tan(60))/(1 - tan(45)tan(60)) =
(1 + sqrt(3))/(1 - (1 * sqrt(3))) = (1 + sqrt(3))/(1 - sqrt(3))

Multiply top and bottom by (1 + sqrt(3)) and you get

((1 + sqrt(3))(1 + sqrt(3)))/((1 - sqrt(3))(1 + sqrt(3)))
(1 + sqrt(3) + sqrt(3) + 3)/(1 + sqrt(3) - sqrt(3) - 3)
(4 + 2sqrt(3))/(1 - 3)
(4 + 2sqrt(3))/(-2)

since both the 4 and (2sqrt(3)) are divided by the -2, you get

-2 - sqrt(3) instead of -2 + sqrt(3), if you were to say tan(105) = -2 - sqrt(3), you get that both sides are equal. However -2 + sqrt(3) gets you -.26795 and tan(105) gets you -3.732




OR




tan(105) = tan(60 + 45)
= [tan(60) + tan(45)] / [1 - tan(60)tan(45)]
= [sqrt(3)+1] / [1-sqrt(3)]

tan(60) = sqrt(3)
tan(45) = 1

Which is the REAL answer??  

Answer
(sqrt(3) + 1)/(1 - sqrt(3)) = (1 + sqrt(3))/(1 - sqrt(3))

In math, we were always told not to leave the sqrt in the denominator, so you get -2 - sqrt(3)

for ex:

if you have sin(45), you don't put the answer as 1/(sqrt(2)), you put the answer as (sqrt(2)/2), am i correct, so there's your proof that you don't leave the sqrt in the denominator.

Here is a website to proove it.

http://faculty.ed.umuc.edu/~swalsh/Math%20Articles/Rationalize.html

why would they have a website that talks about taking out the sqrt in the denominator, if your not suppose to take out the sqrt in the denominator.

So in short even though both answers are correct, mine is the one that is how it is suppose to be writte, also which one do you think is easier to write as the answer.