Question

what is the exact value of tan(195°)?

$\frac{sqrt{3}+1}{1-sqrt{3}}$

$\frac{sqrt{3}-1}{1+sqrt{3}}$

$\frac{sqrt{3}-1}{1+sqrt{3}}$

$\frac{sqrt{3}+1}{1-sqrt{3}}$

Explanation:

Step1: Express \(195^{\circ}\) as a sum of two angles

We know that \(195^{\circ}=150^{\circ}+45^{\circ}\).
The formula for \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\).
Here \(A = 150^{\circ}\) and \(B=45^{\circ}\).
We know that \(\tan150^{\circ}=\tan(180 - 30)^{\circ}=-\tan30^{\circ}=-\frac{\sqrt{3}}{3}\) and \(\tan45^{\circ}=1\).

Step2: Substitute the values into the formula

$$
LATEXBLOCK0
$$

Now, rationalize \(\frac{\sqrt{3}-1}{1 + \sqrt{3}}\):

$$
LATEXBLOCK1
$$

Answer:

\(\frac{\sqrt{3}-1}{1+\sqrt{3}}\) (the second and third options in the given choices are same, and this is the correct one)