# Having square roots in denominator in right angle trig

*apologies ahead of time, i am relativly new to this so I am unsure the proper keystrokes to do the fancy equations on here :| *

I can't remember if this is a general rule or not...

but when solving for $$\sin$$, $$\cos$$, $$\tan$$, etc., if you have a square root (imma abbreviate it as sqrt)in the denominator, are you supposed to rationalize it (multiplying the numerator and denominator by the sqrt value)?

for instance, It's been a while I took a math class, and I could've sworn one of my teachers expressed that you cannot have a sqrt in the denominator when giving your answers for right angle trig. But in this [Precalc crash course] on Youtube (it's the 5 hour one done by UNC@Chapel Hill posted by freeCodeCamp.org, time stamp 01:36:52 - 01:38:52), the professor left them as is.

ex. in the example equation she used, a right angle with a hypotenuse of $$5$$, and an opposite side of $$2$$, solve for sine, cosine, tangent, cosecant, secant, and cotangent. knowing to solve for the last side, adjacent, you use pythagorean theorem, $$a^2 + b^2 = c^2$$, giving you adjacent = $$\sqrt(21)$$.

the professor said that $$\tan = \frac{2}{\sqrt 21}$$ and $$\sec = \frac{5}{\sqrt 21}$$.

isn't the answer though supposed to be $$\frac{2 \sqrt 21}{21}$$ for tangent? $$\frac{5 \sqrt 21}{21}$$ for secant?

help? clarifications?

edit: y'all thx so much for the help and edits : https://www.youtube.com/watch?v=eI4an8aSsgw

• It’s up to you completely. I personally prefer roots in the denominator. Dec 24, 2020 at 19:24
• You almost never "have" to rationalize a denominator, though it does sometimes allow for a simpler form of the equation. I've never been fully sure why schools try to enforce it as absolute fact that you have to, when in university I was told the opposite, that we don't have to. Dec 24, 2020 at 19:25
• It's not that important. Writing $\frac{1}{\sqrt 2}$ or $\frac{\sqrt 2}{2}$ it's not relevant Dec 24, 2020 at 19:25
• It's certainly more convenient to rationalise your denominators when doing calculations by hand. It's not so relevant if you're using a calculator. However, when doing algebra I prefer to do most steps the traditional way, and only use the calculator (if necessary) to get the final result. Dec 24, 2020 at 19:33