Evaluating trigonometric functions

Question

How would I evaluate cot 30 + cot 60? I know that cot 30 = 1/tan30 and cot60 = 1/tan60.

The answer must have a rational denominator where relevant.

I have tried adding them like normal fractions after evaluating, but got the incorrect answer. Any relevant online reading material would be appreciated thanks.

Answer 1

Well, $$\cot 30^\circ=\frac{1}{\tan 30^\circ}=\frac{\cos 30^\circ}{\sin 30^\circ}=\frac{\sqrt{3}/2}{1/2}=\frac{\sqrt{3}}{1}=\sqrt{3}$$ and $$\cot 60^\circ=\frac{1}{\tan 60^\circ}=\frac{\cos 60^\circ}{\sin 60^\circ}=\frac{1/2}{\sqrt{3}/{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$$ (The very last expression in $\cot 60^\circ$ is the result of rationalizing the denominator.)

We see that both $\sqrt{3}$ and $\sqrt{3}/3$ have rational denominators. Can you now add up $\cot 30^\circ$ and $\cot 60^\circ$?

Answer 2

$$\cot \theta = \frac{\cos \theta}{\sin \theta}$$

You should know sine and cosine of $30^\circ$ and $60^\circ$.

Answer 3

Well tan(a)=sin(a)/cos(a) So, you want cos(30)/sin(30) + cos(60)/sin(60) = $\frac{\sqrt{3}/2}{1/2} + \frac{1/2}{\sqrt{3}/2} = \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}}+\frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} = \frac{4 * \sqrt{3}}{3}$