Question

tan^-1(sqrt3)/3
In +2 pi n notation as well, please?
*see screenshot of question below

Answer 1

http://openstudy.com/study#/updates/521510aae4b0450ed75e3401

Answer 2

Is this the same thing as +2 pi n notation? I am doing an online class and I don't have a teacher for this material--I'm sorry if that's a silly question

Answer 3

can you give us a screenshot of the material?

Answer 4

Are you evaluating the expression? or solving an equation? the "+ 2pi n" sounds like you are solving an equation, but that isn't what you presented above. Is that what you meant?

Answer 5

I uploaded a screenshot of the question

Answer 6

Oh, Lordy Lordy. There should be no need for any +2npi for this problem. It's an inverse FUNCTION. it has ONE OUTPUT. I don't know why your teacher would ask it like that.

The range of inverse tangent is -pi/2<y<pi/2

That's how it gets to be a called a "function".

Answer 7

Haha it is very confusing to me as well! So is the answer -pi/2 < y < pi/2 ?

Answer 8

\[\let x=\tan^{-1} \frac{ \sqrt{3} }{ 3}\]
\[\tan x=\frac{ \sqrt{3} }{3 }=\frac{ 1 }{\sqrt{3} }=\frac{ \frac{ 1 }{2 } }{\frac{ \sqrt{3 } }{ 2 } }=\tan \frac{ \pi }{6 }= \tan \left( n \pi+\frac{ \pi }{6 } \right)\]
\[x=n \pi+\frac{ \pi }{6 }\]

Answer 9

See.... asking the function value:\[\large \tan^{-1} \left( \frac{ \sqrt{3} }{ 3 } \right)=?\]
is a different question than asking solve the equation:\[\large tanx= \left( \frac{ \sqrt{3} }{ 3 } \right)\]In the 2nd case, solving the equation, you USE the inverse function but then you need the +2kpi (I call it the "incrementor" but that's my own word, lol) to capture all the coterminal solutions. But those are SOLUTIONS TO AN EQUATION and not the OUTPUT OF A FUNCTION.

If you look up the definition of the inverse tangent function, er, anywhere, you will see that it has the limited range. It HAS to, otherwise it is NOT a function because a function can only have ONE OUTPUT PER INPUT. That's the rule for functions!! So I'm a bit disturbed that your teacher is asking the question this way. Wish I could debate this with him/her. lol :)

Answer 10

Here is another question from the previous attempt at this assignment:
Is the same also true here?

Answer 11

Oh goodness yes. Your answer was right. \[\large \cos^{-1} \left( \frac{ 1 }{ 2 } \right)=\frac{ \pi }{3 }\]

it is NOT TRUE that \[\large \cos^{-1} \left( \frac{ 1 }{ 2 } \right)=\frac{ 5\pi }{3 }\] because that's in quadrant 4 which is not in the RANGE of the inverse cosine. It's just wrong. I understand that it's what your teacher was looking for, but it's wrong as the question is asked. If the question asked:

What set includes all x such that cos(x)=1/2?

....then it would be correct to include the 4th quadrant angle and also the "incrementor" (the +2kpi). But that's a different question than "what is the inverse function value"?

Answer 12

It's analogous to this: if I asked you to evaluate:\[\large \sqrt{9}=?\]you SHOULD tell me \[\large \sqrt{9}=3\] because the square root FUNCTION (again, key that it is a function) is DEFINED to mean the positive (or "principal") square root.

Now if I ask you to SOLVE:\[\large x^2=9\]then you should give the solution set {3,-3} because BOTH are solutions to the EQUATION.

Equations may have multiple solutions. Functions MAY NOT have multiple outputs for the same input.

Answer 13

Oh! Well that explains why I have redone it 4 or 5 times. :/ So if I were to solve the problem like my instructor wants it, I should consider arctan=(sqrt3)/3 as an equation and not a function.

Answer 14

It's so frustrating to have to redo this unit so many times! The book teaches it the way you have explained it, yet, the teacher marks the answers wrong without any real answer as to why.

Answer 15

Well, I would not say to "consider" it that way. It's not an equation. But you know what the instructor is looking for. The key thing then to remember is that for tangent (and cotangent) the period of the function is PI, so the incrementor is kpi. If you do any for sine, cosine, csc or sec, the period is 2pi and you need 2 different solutions. But yeah, they seem to want you to TREAT it like it's an equation, even though that isn't what the problem says!

Answer 16

Ok. Thank you for answering my question! This is really helpful

Answer 17

No problem.... it frustrates me to see these topics that I hold dear being taught in an imprecise way, lol. I always say, "math is precise!" :)

This might help: http://www.dummies.com/how-to/content/identify-the-domains-and-ranges-of-inverse-trigono.html
Maybe you can even set your teacher straight! ;) (But don't make them mad, lol!)