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,

sin75° = sin45° cos30° + cos45 °sin30° =

___ Ö3/8

+

___ Ö1/8

Perhaps the high school students don't know that Ptolemy discovered that

sin36° =

Ö

5/8-Ö5/8

. 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

sin54°=

Ö

3/8+Ö5/8

.

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

Ö

3/8+Ö5/8

is not fully simplified. It can be further simplified as

____ Ö5/16

+1/4

, or even more simply,

(Ö5+1)/4, which Ptolemy would tell you is just half the golden ratio.

But how is a person to recognize that

Ö

3/8+Ö5/8

is not fully simplified?

The reason I ask is that I calculated exact representations of sin84^° - - one is

æ Ö

9/32+   __ Ö45   /32

+

Ö

5/32-Ö5/32

and the other is

æ  ú Ö

7/16+Ö5/16+   æ Ö 15/128+   __ Ö45   /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 sin3^°, I use the half angle formula, which gives me

æ  ú Ö

1/2-   æ Ö 9/128+   __ Ö45   /128   - Ö   5/128-Ö5/128

I checked my work, and this is, in fact, an exact representation of sin3^°, 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

=

__ Ö15

/8+Ö3/8+

Ö

5/32-Ö5/32

If both are true (or at least if they are equal to one another) then

__ Ö15

/8+Ö3/8=

æ Ö

9/32+   __ Ö45   /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

_________ Öleftside2

=rightside

So let's square

__ Ö15

/8+Ö3/8

, then take its square root:

æ Ö

(   __ Ö15   /8+Ö3/8)2

=

æ Ö

15/64+2   __ Ö45   /64+3/64

=

æ Ö

9/32+   __ Ö45   /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+

__ Ö45

/32

is a ''perfect square'' - - the square of

__ Ö15

/8+Ö3/8

. 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

(Öa+Öb)2 = a+b+2

__ Öa b

, 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

__ Öa b

.

Applying this, if

9/32+

__ Ö45

/32

is a perfect square, then

9/32+

__ Ö45

/32 = a+b+2

__ Öa b

That means

__ Ö45

/32 = 2

__ Öa b

So

__ Ö45

/64 =

__ Öa b

So 45/4096 = a b

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

____ Ö3/64

+

_____ Ö15/64

is

9/32+

__ Ö45

/32

- - - -

Let's see if

Ö

5/32-Ö5/32

can be simplified this way. If the expression inside the

Ö

is a perfect square then

5/32-Ö5/32 = (Öa-Öb)²

5/32-Ö5/32 = a+b-2

__ Öa b

So a+b=5/32

and 4a b=5/1024

Alas, a and b are not rational numbers;

a=(5+2Ö5)/64, and b=(5-2Ö5)/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