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Exact Values of Trigonometric Functions
By Graeme Mcrae on Sunday, December 30, 2001 - 09:30 am:

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.

As every high school student learns, sin 30º=1/2, and sin 45º=Ö(1/2), etc.

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

For example, sin 75º = sin 45º cos 30º + cos 45º sin 30º = Ö(3/8)+Ö(1/8)

Perhaps the high school students don't know that Ptolemy discovered that sin 36º = 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 sin 54º=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

Ö(5/16)+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 sin 84º -- one is

Ö(9/32+Ö(45)/32)+Ö(5/32-Ö5/32)

and the other is

Ö(7/16+Ö5/16+Ö(15/128+Ö(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 3º, I use the half angle formula, which gives me

Ö(1/2-Ö(9/128+Ö(45)/128)-Ö(5/128-Ö5/128))

I checked my work, and this is, in fact, an exact representation of sin 3º, but I have no way of knowing if a simpler representation exists.

Can anyone help me? Thanks in advance.

By Yatir Halevi on Sunday, December 30, 2001 - 12:42 pm:

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:
Ö(5/32 - Ö5/32) + Ö(15)/8 + Ö(3)/8

sin3 to:
Ö(- Ö(5/128 - Ö5/128) - Ö(15)/16 - Ö(3)/16 + 1/2)

Yatir

By Graeme Mcrae on Sunday, December 30, 2001 - 06:13 pm:

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

my sin84 = Ö(9/32+Ö(45)/32)+Ö(5/32-Ö5/32)
your sin84 = Ö(15)/8+Ö(3)/8+Ö(5/32-Ö5/32)

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

Ö(15)/8+Ö(3)/8 = Ö(9/32+Ö(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

Ö(leftside^2) = rightside

So let's square Ö(15)/8+Ö(3)/8, then take its square root:

Ö((Ö(15)/8+Ö(3)/8)^2) =
Ö(15/64 + 2Ö(45)/64 + 3/64) =
Ö(9/32 + Ö(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+Ö(45)/32 is a "perfect square" -- the square of Ö(15)/8+Ö(3)/8. To simplify the Ö 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

(Ö(a)+Ö(b))2= a+b+2Ö(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 "separated" so that the rational part is a+b, and the real part is 2Ö(ab).

Applying this, if 9/32+Ö(45)/32 is a perfect square, then

9/32+Ö(45)/32 = a+b+2Ö(ab)

That means Ö(45)/32 = 2Ö(ab)
So Ö(45)/64 = Ö(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 Ö(3/64)+Ö(15/64)
is 9/32+Ö(45)/32

----

Let's see if Ö(5/32-Ö5/32) can be simpified this way. If the expression inside the Ö is a perfect square then

5/32-Ö5/32 = (Ö(a)-Ö(b))2
5/32-Ö5/32 = a+b-2Ö(ab)
So a+b=5/32
and 4ab=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.

By Yatir Halevi on Sunday, December 30, 2001 - 08:42 pm:

You Welcome,
Hope it helped.

Yatir