Originally Posted by
Reckoner I never meant to discourage you! I only wanted to nudge you in the right direction. Keep in mind that there are plenty of people that have difficulty grasping basic algebra, or even simple arithmetic, and you seem to at least know what you are doing there if you are doing trigonometry. Even the greatest mathematicians face difficulties, and yours can be overcome as long as you put in the necessary effort.
Now, I wouldn't expect you to "see" how I made that jump because I skipped a bunch of steps in-between. But the idea was that you could get to the same answer if you used the same starting point. Here is what I did:
$\displaystyle a^2 = 10000^2 + 25000^2\tan^222^\circ$
$\displaystyle \Rightarrow a^2 = 100000000 + 625000000\tan^222^\circ$ (Evaluating exponents)
$\displaystyle \Rightarrow a = \sqrt{100000000 + 625000000\tan^222^\circ}$ (Taking square root of both sides; we can ignore the negative root since $\displaystyle a > 0$)
$\displaystyle \Rightarrow a = \sqrt{4\cdot25000000 + 25\cdot25000000\tan^222^\circ}$ (Factoring)
$\displaystyle \Rightarrow a = \sqrt{25000000\left(4 + 25\tan^222^\circ\right)}$ (Factoring)
$\displaystyle \Rightarrow a = \sqrt{25000000}\sqrt{4 + 25\tan^222^\circ}$ (Splitting square root: $\displaystyle \sqrt{ab}=\sqrt a\sqrt b$)
$\displaystyle \Rightarrow a = 5000\sqrt{4 + 25\tan^222^\circ}$ (Evaluating: $\displaystyle \sqrt{25000000} = \sqrt{25\cdot10^6} = \sqrt{25}\sqrt{10^6} = 5\cdot10^3 = 5000$)
And of course, if you are just looking for an approximation, you could have stopped at the first step and threw it into the calculator. I just like keeping things in their exact value.