*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][1] 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 [1]: https://www.youtube.com/watch?v=eI4an8aSsgw

  • $\begingroup$ It’s up to you completely. I personally prefer roots in the denominator. $\endgroup$
    – Tavish
    Dec 24, 2020 at 19:24
  • 1
    $\begingroup$ 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. $\endgroup$ Dec 24, 2020 at 19:25
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    $\begingroup$ It's not that important. Writing $\frac{1}{\sqrt 2}$ or $\frac{\sqrt 2}{2}$ it's not relevant $\endgroup$
    – Raffaele
    Dec 24, 2020 at 19:25
  • $\begingroup$ 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. $\endgroup$
    – PM 2Ring
    Dec 24, 2020 at 19:33

1 Answer 1


It's only subjective. Most maths teachers find it ugly to put a square root at the denominator, but there is nothing false in doing it.

  • $\begingroup$ Do you have any support for what you say about "most mathematicians" ? $\endgroup$
    – coffeemath
    Dec 24, 2020 at 19:32
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    $\begingroup$ Sorry I wanted to say "most math teachers" : in France it is something officially taught. I edited. $\endgroup$
    – math
    Dec 24, 2020 at 19:35

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