# 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.