Need square root help

[Kage2021]

Ok. This question might not belong here but since it derived from a trig question I felt it would slide by. I would like to say the reason I dont know how to do this is that some math teacher along the line failed me before college but its probably more appropriate to say that I failed them.

Regardless here is an example of my problem.
(without calculator)

3Cos30(deg) - 5Sin30(deg) =

So of course from special angles I know that the cos of 30 degrees is equal to the square root of 3 over two and the sin is equal to the square root of 1 over two. Which leaves me with something to the effect of

3 * (square root of) 3/2 - 5 * (square root of) 1/2

So how do I multiply 3 times the square root of 3 without a calculator and when i'm finished, how do I rationlize such a number? Apparently there are some rules for these types of problems but I have scourerd the internet (for about an hour) and havent seen anything. Any help or direction to move would be great, thanks.

 

[galactus]

\(\displaystyle 3cos(30)=\frac{3\sqrt{3}}{2}\)

\(\displaystyle 5sin(30)=\frac{5}{2}\)


\(\displaystyle 3cos(30)-5sin(30)=\frac{3\sqrt{3}}{2}-\frac{5}{2}=\frac{3\sqrt{3}-5}{2}\)

 

[Kage2021]

Ok good any idea where I can find some rules governing this sort of thing so that I can apply it to all of these sort of problems as opposed to asking a million questions?

 

[Kage2021]

for example
how does 2*(squarerootof)3 all over 3 + 4*(square root of)2 all over 2

"rationalize" to 2*(square root of)3 + 6*(square root of)2 all over 3? Dont you have to have a common denominator to "simplify" any further? Where the **** is three coming from? This is confusing the **** out of me as well as starting to get on my nerves.

 

[galactus]

You mean:

\(\displaystyle \L\\\frac{2\sqrt{3}}{3}+\frac{4\sqrt{2}}{2}\).....[1]

You know it's equal to:

\(\displaystyle \L\\\frac{2\sqrt{3}+6\sqrt{2}}{3}\)....[2]

But how?.


Multiply the top and bottom of the right side of [1] by 3/2:

\(\displaystyle \L\\\frac{2\sqrt{3}}{3}+\frac{4\sqrt{2}\cdot\frac{3}{2}}{2\cdot{\frac{3}{2}}\)

Is this what you mean?.

You get [2]. See?. It's always those little things that we miss.