How to Solve Tan(x) = 3 Without Calculator: Trigonometry Question

[Constantinos]

This is no homework exercise, just plain curiosity for an exercise, so give it lower priority.

Homework Statement

Is it possible to solve tan(x) = 3 without the use of a calculator?

The knowns:
1) It originaly came from trying to solve 1/(cos^2) -2tan = 4.
1 = cos^2+sin^2 so this can be converted into tan^2 -2tan-3 = 0 which solves for
tan = -1(easy) and tan = 3(?)
If there is another way to solve without tan=3 then it would be ok, but still more interesting if tanx = 3 can be solved.
2) All sin,cos,tan,cot for angles 0,30,45,60,90 and multiples.
3) Trigonometric circle
All else has to be proved.

Homework Equations

The Attempt at a Solution

I tried using the trigonometric circle, but I couldn't find any useful relation. Tried squaring, nothing happens. Tried converting to sin and cos, nothing. I know the solution is 2pi/5 but can't find a way to bring it up to the rhs of the equation.

I just thing that the textbook is juts wrong to ask that with this kind of information.

 

[QuarkCharmer]

Without using an expanded unit circle from 0 to 2pi in steps of pi/5, I don't see how it would be possible. Someone correct me if I am just not seeing it. Are you sure you have to solve without a calculator? Does that rule out a slide rule? :)

 

[Filip Larsen]

I know the solution is 2pi/5 but can't find a way to bring it up to the rhs of the equation.

I must concur with QuarkCharmer that 2/5 pi is not a solution. See for instance [1].

[1] http://www.wolframalpha.com/input/?i=tan(2+pi+/+5)

 

[Dick]

I must concur with QuarkCharmer that 2/5 pi is not a solution. See for instance [1].

[1] http://www.wolframalpha.com/input/?i=tan(2+pi+/+5)

Right. tan(2*pi/5) is only approximately 3. It's more like 3.077683537175253.

 

[Constantinos]

Well the solution is written in the back of the textbook. I guess I should have checked it first silly me!

Well all the other exercises of the textbook don't require a calculator (and are solved pretty easily) and it is written in the introduction of the book that none is needed. So assumed so.

But since the solution offered is approximate, I also think that there is no way this could be solved without a calculator. Perhaps a mistake of the book then.

Thanks for the replies people!

 

[olivermsun]

Hmm. I don't know about your teacher but mine probably would have accepted arctan(3) as an answer if it was a "non-calculator" problem.