So I am asked to use the appropriate compound angle formula to create an equivalent expression for tan(-15). my solution was: \begin{align}\tan(-15) &= \tan(30-45) \\ &=\frac{\tan30 - \tan45}{ 1+\tan30 \cdot \tan45} \\ &= \frac{(\frac{\sqrt{3}}{3}) - 1}{ 1 + (\frac{\sqrt3}{3})} \\ &=\frac{\sqrt3 - 3}{3 + \sqrt3} \\ &=\frac{-12 + 6\sqrt3}{6} \\ &=-2 + \sqrt3 \end{align}
However, the teacher's solution was
\begin{align}\tan(-15) &= -\tan(15) \\ &=\tan(30/2) \\ &=\sqrt{\frac{(1 - sqrt3/2)}{(1 + sqrt3/2)}} \\ &=\sqrt{7 - 4\sqrt3}\\ &=-\sqrt{7 - 4\sqrt3} \\ &=-(2 - \sqrt3) \\ &=-2 + \sqrt3 \\ \end{align}
My teacher's solution used the related angle and half angle formulas. is my solution to the answer considered wrong? how do i know when to use the corelated and related angles?
I have no idea when to use these formulas and ive been using my method for the past hour. im afraid ive been doing it wrong the whole time
Your solution is fine. Note that the value agrees with the teacher's. There are a number of routes to the solution, of which you have shown two.