Trigonometric functions: express sin(x) in terms of tan(x)

[Mentallic]

Homework Statement

I want to express sin(x) in terms of tan(x).

Homework Equations

tan(x)=sin(x)/cos(x)
1+tan²(x)=sec²(x)

The Attempt at a Solution

sin(x)=cos(x)tan(x)
At this point I realize this is assuming $x\neq \pi/2+k\pi$

$$cos^2(x)=\frac{1}{1+tan^2(x)}$$

therefore, $$sin(x)=\frac{tan(x)}{\sqrt{1+tan^2(x)}}$$

But I graphed this and it only looks right half of the time. What I should have is the plus or minus when taking the root of cos²(x) but I need the plus half the time, and the minus the other half of the time and then merge them to describe sin(x) in terms of tan(x).

How should I go about this problem and realize what must be done without the use of graphing tools.

 

[LCKurtz]

Of course you have pinpointed the problem, that being the necessity to have the appropriate sign on the square root. If you multiply your expression by sgn(cos(x)), all will be well except that you may object to having that cosine in your formula. I don't think there is any way to express the sign just in terms of the tangent function because the tangent function has two periods for one for the cosine and sine. You might be able to concoct a formula using greatest integer and mod functions to make a square wave to multiply it by that doesn't involve the cosine, but that is even less satisfactory.

 

[Bohrok]

You should have taken a look here first to find expressing sine with tangent
http://en.wikipedia.org/wiki/Trig_identities#Related_identities

 

[Mentallic]

Of course you have pinpointed the problem, that being the necessity to have the appropriate sign on the square root. If you multiply your expression by sgn(cos(x)), all will be well except that you may object to having that cosine in your formula. I don't think there is any way to express the sign just in terms of the tangent function because the tangent function has two periods for one for the cosine and sine. You might be able to concoct a formula using greatest integer and mod functions to make a square wave to multiply it by that doesn't involve the cosine, but that is even less satisfactory.

I'd probably have the most luck concocting a piecewise formula for this case since that is what I have experience in doing. By the way, what is sgn?

You should have taken a look here first to find expressing sine with tangent
http://en.wikipedia.org/wiki/Trig_identities#Related_identities

The formula there gives $$sin(x)=\pm\frac{tanx}{1+tan^2x}$$ so by being as brief as possible, this is the best that could be done. They don't mention for what domain it is plus and where is it minus.

 

[LCKurtz]

I'd probably have the most luck concocting a piecewise formula for this case since that is what I have experience in doing. By the way, what is sgn?

sgn is sometimes used as an abbreviation of the signum or "sign" function.

sgn(x) = 1 if x > 0 and -1 if x < 0

That's why sgn(cos(x)) multiplies your answer by the appropriate choice of + or -.

See http://en.wikipedia.org/wiki/Sign_function