solving trig equation Tan2 (x) + 3 = 0 for all solutions between 0 and 360 degrees

[simonw144]

Hi All

I am struggling with the following question I have been given and am looking for some guidance on where to go next. I have been asked to solve the following for all angles between 0 and 360 degrees

Tan² (x) + 3 = 0

So far

I have got it to

Tan (x) = square root of negative 3

and have said it can not be solved as you can not have a square root of negative 3.

I have been told I need a more complete answer but am not sure where to go next.

We have done complex numbers in the same class so I can solve the square root of negative 3 to 1.73i.

Can I now use the 1.73i with it just being a case of finding the arctan of 1.73 to find the angles or am I looking at this in the wrong way.

Previously in the complex numbers I have had a real and imaginary part but in this I only have the imaginary part.

Any guidance would be gratefully received as I do not really know what to do next and all the research I have done has lead nowhere

 

[tkhunny]

How would you solve this?

x^2 - 1 = 0

TWO solutions, right? Do EXACTLY the same thing and this might have some potential to give what you are expecting.

However, does this actually work? Can you get Complex Values from a standard REAL Tangent Function? You probably should think about this before proceeding.

In addition, how are you going to know if your COMPLEX results are between 0 and 2*pi? Do you mean just the REAL part or something else?

 

[Dr.Peterson]

I have been asked to solve the following for all angles between 0 and 360 degrees

Tan² (x) + 3 = 0

So far

I have got it to

Tan (x) = square root of negative 3

and have said it can not be solved as you can not have a square root of negative 3.

I have been told I need a more complete answer but am not sure where to go next.

We have done complex numbers in the same class so I can solve the square root of negative 3 to 1.73i.

Can I now use the 1.73i with it just being a case of finding the arctan of 1.73 to find the angles or am I looking at this in the wrong way.

Previously in the complex numbers I have had a real and imaginary part but in this I only have the imaginary part.

Any guidance would be gratefully received as I do not really know what to do next and all the research I have done has lead nowhere

As I read the question, x has to be an ANGLE, in particular a REAL NUMBER between 0 and 360 (degrees).

So imaginary solutions are not allowed. The answer is that there is NO SOLUTION.

Can you tell us more about what you were told? Did you quote the problem exactly, or is there something you left out?

 

[tkhunny]

As I read the question, x has to be an ANGLE, in particular a REAL NUMBER between 0 and 360 (degrees).

So imaginary solutions are not allowed. The answer is that there is NO SOLUTION.

Agreed This is a very important point. I my response, I just ignored it as particularly inconvenient to any sort of solution.

 

[simonw144]

Hi all

Thank you for the help, my answer was that the problem was not solvable as there is no answer for the square root of -3. The feedback I got was that I needed to submit a more definite/complete solution. The feedback did not say my answer was wrong but that it needed to be more complete.

So I think that maybe i should say there is an answer to the square root of negative 3 but it is an imaginary number and as such there is no solution as I can not use the imaginary number with Tan.

 

[Dr.Peterson]

Thank you for the help, my answer was that the problem was not solvable as there is no answer for the square root of -3. The feedback I got was that I needed to submit a more definite/complete solution. The feedback did not say my answer was wrong but that it needed to be more complete.

So I think that maybe i should say there is an answer to the square root of negative 3 but it is an imaginary number and as such there is no solution as I can not use the imaginary number with Tan.

It is actually possible to extend the tangent function so that its domain and range include complex numbers; so what you say is not technically true. The point to emphasize is that the context excludes complex numbers, not that the tan function does so in itself. Similarly, you should not say that there is no "answer" for $\displaystyle \sqrt{-3}$, but that it is not a real number.

Beyond that, I'd need to know more of your context to get a sense of what kind of "completeness" is required.

 

[simonw144]

Again Thank you for your help. As you may of guessed maths is not my strength.

Basically where I am now is I have

Tan (x) = 1.73i

Going by your previous advice can I now use the 1.73i to get an angle as I am not sure how you would do this.

Tan^-1 1.73 = x would give me 59.9 degrees. Would this be 59.9i degrees or am i totally missing this