Trigonometry (Double angle formula)

1. Apr 11, 2013

trollcast

1. The problem statement, all variables and given/known data
Calculate the value of $\tan(\frac{\pi}{12})$ without the use of a calculator and showing all steps.

2. Relevant equations

Compound angle formula / Double angle formula

3. The attempt at a solution

Using an equilateral triangle of side length 2 I've shown that:
$$\tan(\frac{\pi}{6})=\frac{1}{\sqrt{3}}$$
Then using the double angle formula:
$$\tan(2\theta)=\frac{2\tan(\theta)}{1-\tan^2(\theta)}$$
Substituting the value for $\tan(\frac{\pi}{12})$ in as 2θ and solving for $\tan(\theta)$:
$$\tan(\frac{\pi}{12})=\pm2-\sqrt(3)$$

However how do I show that I only need to take the positive solution?

Thanks

2. Apr 11, 2013

micromass

Staff Emeritus
You know that $\pi/12$ is in the first quadrant. What must the tangent be of an angle in the first quadrant? Positive or negative?

3. Apr 11, 2013

trollcast

Thanks micro don't know how I missed that.

So the tangent should be positive since its in the first quadrant, therefore the negative solution is not possible?

4. Apr 11, 2013

micromass

Staff Emeritus
That's it!

5. Apr 11, 2013

trollcast

Thanks again, I just actually realised I could have done it by finding $\tan(\frac{\pi}{4})$ and then using the compound angle formula to calculate $\tan(\frac{\pi}{4}-\frac{\pi}{6})$ , which would only give one solution at the end.