Expert: Jack Cheng - 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
Hi Karen,

They (We) are both right.

I just left mine as [sqrt(3)+1] / [1-sqrt(3)] while Sherman multiplied it by the conjugate and reduced it further. You get the same answer eventually though, which I assume is -3.732.

I do admit that my answer is somewhat confusing at the end (with the tan(60)..). I apologize for it.

~ Jack