I need some help.

I'm making a page for my website, mcraefamily.com/MathHelp/GeometrySpecialAngles.htm , which gives the exact values of the sine of various angles.

Also, every high school student learns the formulas for sines of sums of angles.

For example, $\sin {75}^{\circ }=\sin {45}^{\circ }\cos {30}^{\circ }+\cos {45}^{\circ }\sin {30}^{\circ }=\sqrt{3/8}+\sqrt{1/8}$

Perhaps the high school students don't know that Ptolemy discovered that $\sin {36}^{\circ }=\sqrt{5/8-\sqrt{5}/8}$. Of course, he didn't express it in those terms; instead he invoked the golden ratio, but in modern terms, this is what he discovered.

Using this result, and the half-angle formula, a rich selection of exact trig functions can be represented.

For example, with a little calculation it is easy to discover that $\sin {54}^{\circ }=\sqrt{3/8+\sqrt{5}/8}$.

I'm getting to the part where I need help...

$\sqrt{3/8+\sqrt{5}/8}$ is not fully simplified. It can be further simplified as

$\sqrt{5/16}+1/4$, or even more simply,

$(\sqrt{5}+1)/4$, which Ptolemy would tell you is just half the golden ratio.

But how is a person to recognize that $\sqrt{3/8+\sqrt{5}/8}$ is not fully simplified?

The reason I ask is that I calculated exact representations of $\sin {84}^{\circ }$ - - one is

$\sqrt{9/32+\sqrt{45}/32}+\sqrt{5/32-\sqrt{5}/32}$

and the other is

$\sqrt{7/16+\sqrt{5}/16+\sqrt{15/128+\sqrt{45}/128}}$

Both are quite correct; the first looks a bit simpler than the second; however I fear there may be lurking just beyond my reach a far simpler representation of this same value.

It gets worse.

To find $\sin 3^{\circ }$, I use the half angle formula, which gives me

$\sqrt{1/2-\sqrt{9/128+\sqrt{45}/128}-\sqrt{5/128-\sqrt{5}/128}}$ I checked my work, and this is, in fact, an exact representation of $\sin 3^{\circ }$, but I have no way of knowing if a simpler representation exists.
Can anyone help me? Thanks in advance.

You might want to accept it or not (I know I wouldn't), but using my computer i simplified (So, I have no proof)
sin84 to:

Thank you, Yatir.

You're right, I am reluctant to take such things on faith, or on the authority of others. But luckily, I can verify that the formulas you provided are in perfect agreement with the (more complex) formulas I derived on my own. Moreover, the simplification you provided gives me insight into a possible method of simplifying such expressions.

In this case, I calculated

your sin84 $=\sqrt{15}/8+\sqrt{3}/8+\sqrt{5/32-\sqrt{5}/32}$
If both are true (or at least if they are equal to one another) then

$\sqrt{15}/8+\sqrt{3}/8=\sqrt{9/32+\sqrt{45}/32}$
The best way to verify this is to square the left side, then take its square root.

In other words, if

leftside = rightside

then

$\sqrt{{\mathrm{leftside}}^2}=\mathrm{rightside}$

So let's square $\sqrt{15}/8+\sqrt{3}/8$, then take its square root:

$\sqrt{(\sqrt{15}/8+\sqrt{3}/8{)}^2}=$

$\sqrt{15/64+2\sqrt{45}/64+3/64}=$

$\sqrt{9/32+\sqrt{45}/32}$
Once you show me the simplified expression, I can use this method to verify that it is identical to my needlessly complex expression.

However, I still find it difficult to simplify such expressions. Looking back on this one, I can see how it could be done...

Clearly, in this case, $9/32+\sqrt{45}/32$ is a ''perfect square'' - - the square of $\sqrt{15}/8+\sqrt{3}/8$. To simplify the of the former expression, I need a method to recognize such perfect squares. Not just a method, but a constructive method: I would like to find its square root.
Toward that end I see that

$(\sqrt{a}+\sqrt{b}{)}^2=a+b+2\sqrt{ab}$, where $a$ and $b$ are rational numbers. At this point, I conjecture that the rational and irrational parts of such a sum can be ßeparated" so that the rational part is $a+b$, and the real part is $2\sqrt{ab}$.

Applying this, if $9/32+\sqrt{45}/32$ is a perfect square, then

$9/32+\sqrt{45}/32=a+b+2\sqrt{ab}$

That means $\sqrt{45}/32=2\sqrt{ab}$

So $\sqrt{45}/64=\sqrt{ab}$

So $45/4096=ab$

And $a+b=9/32$

Solving these two equations for $a$ and $b$, I get

$a=3/64$, and $b=15/64$ (or vice versa)

And sure enough, the square of $\sqrt{3/64}+\sqrt{15/64}$

is $9/32+\sqrt{45}/32$

- - - -

Let's see if $\sqrt{5/32-\sqrt{5}/32}$ can be simplified this way. If the expression inside the $\sqrt{}$ is a perfect square then

$5/32-\sqrt{5}/32=(\sqrt{a}-\sqrt{b}{)}^2$

$5/32-\sqrt{5}/32=a+b-2\sqrt{ab}$

So $a+b=5/32$

and $4ab=5/1024$

Alas, $a$ and $b$ are not rational numbers;

$a=(5+2\sqrt{5})/64$, and $b=(5-2\sqrt{5})/64$ (or vice versa).
So I can't use this method to further simplify sin84.

Again, thanks for your help, I really appreciate it.

You Welcome,
Hope it helped.

Yatir