*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
Best Answer
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.