Question

determining the exact trig function value of an angle
what is the exact value of $\tan(105^{\circ})$?
options: $-2 + \sqrt{3}$, $2 - \sqrt{3}$, $-2 - \sqrt{3}$, $2 + \sqrt{3}$ (with $2 + \sqrt{3}$ marked incorrect)

Explanation:

Step1: Express 105° as a sum

We know that \(105^{\circ}=60^{\circ} + 45^{\circ}\). The tangent addition formula is \(\tan(A + B)=\frac{\tan A+\tan B}{1 - \tan A\tan B}\).

Step2: Substitute A = 60° and B = 45°

We know that \(\tan60^{\circ}=\sqrt{3}\) and \(\tan45^{\circ}=1\). Substituting these values into the formula:
\[

$$\begin{align*} \tan(105^{\circ})&=\tan(60^{\circ}+ 45^{\circ})\\ &=\frac{\tan60^{\circ}+\tan45^{\circ}}{1-\tan60^{\circ}\tan45^{\circ}}\\ &=\frac{\sqrt{3}+ 1}{1-\sqrt{3}\times1}\\ \end{align*}$$

\]

Step3: Rationalize the denominator

Multiply the numerator and denominator by \(1 + \sqrt{3}\):
\[

$$\begin{align*} \frac{\sqrt{3}+1}{1 - \sqrt{3}}\times\frac{1+\sqrt{3}}{1+\sqrt{3}}&=\frac{(\sqrt{3}+1)(1 + \sqrt{3})}{(1)^2-(\sqrt{3})^2}\\ &=\frac{(\sqrt{3}+1)^2}{1 - 3}\\ &=\frac{3 + 2\sqrt{3}+1}{-2}\\ &=\frac{4+2\sqrt{3}}{-2}\\ &=-2-\sqrt{3} \end{align*}$$

\]

Answer:

\(-2-\sqrt{3}\)