A pretty difficult trig identity

Originally posted by kbaumen
I had this as homework in my algebra course and it took a fair amount of time for me to find the solution. It's actually quite difficult to notice.

Prove the identity:
cos(2pi/7) + cos(4pi/7) + cos(6pi/7) = -1/2

(cos(2pi/7)) (1+2+3) = -1/2

(cos(2pi/7))(6) = -1/2

cos(2pi/7) = -1/12

~.62 =/= -1/12

Originally posted by eldragonfly
(cos(2pi/7)) (1+2+3) = -1/2

(cos(2pi/7))(6) = -1/2

cos(2pi/7) = -1/12

~.62 =/= -1/12

Ermm, not quite. That's a violation of basic trigonometry rules.

cos2x =/= 2cosx.

A mistake students often make.

gotcha.

Originally posted by FabianFnas
As computers are using binary memories, 1/3 can't be expressed exactly.
However, a computer with trinary memories (where a bit can store one out of three states), 1/3 is perfectly possible to store (as 1/3 is 0.1 trinary).
But this trinary computer can't prove the (1/2)*2=1 identity as 1/2 cannot be exactly represented in trinary memories.

I'm not talking about a computer. I'm talking about one of these:

http://en.wikipedia.org/wiki/Finite_state_machine

You have finitely many states, but my challenge is to construct an FSA which returns a language of strings which converge to some desired number, like 1/3.

Originally posted by mtthw
Well, you can buy calculators easily enough that can accurately do fractional arithmetic - they just don't store them that way. You can accurately store a fraction as two integers - numerator and denominator.

Then what will be the answer of this question? (1/pi)*pi do you think, concidering that pi cannot be stored exactly?

If you get the answer =1, then see what will happen if you subtract 1 from it. Perhaps your calculater just round it to =1.

Originally posted by FabianFnas
Then what will be the answer of this question? (1/pi)*pi do you think, concidering that pi cannot be stored exactly?

If you get the answer =1, then see what will happen if you subtract 1 from it. Perhaps your calculater just round it to =1.

Now you've pointed to a drawback (at least for my Citizen). Numerator or denominator can only be integers. But they are stored in memory also in that way - as two integers.

Originally posted by kbaumen
Now you've pointed to a drawback (at least for my Citizen). Numerator or denominator can only be integers. But they are stored in memory also in that way - as two integers.

What happens if you put in the expression 1/3 and hit the = button. You see 0.333... at the window. How is it stored? As 1/3 (the fractional number with a integer numerator and denominator)? Or the decimal representation of 1/3 with as many decimals as the memory can hold?

I think you have to put your calculator in some sort of fractional mode in order to calculate fractions exactly. Am I right in this?

My teacher said once that pi is not an exact value unless you know all of its decimals, and as there are infinite number of decimals, pi cannot be exact by itself. He showed me himself with a calculator: HE entered pi with as many decimals that the window could hold. Then he subtracted this with the calculators internal constant of pi and Voilà, the result was not zero. "You see? pi is not an exact value!"

I didn't believe him then, but I had difficulties to explain to him his erratic explanation. After I took the class of 'numerical methods' I could explain it with precision, but at this time he was retired and didn't give a rats *ss anymore.

Originally posted by FabianFnas
Then what will be the answer of this question? (1/pi)*pi do you think, concidering that pi cannot be stored exactly?

If you get the answer =1, then see what will happen if you subtract 1 from it. Perhaps your calculater just round it to =1.

True, but an application such as MathCad that can do symbolic manipulation can handle that with no difficulties.

Originally posted by FabianFnas
My teacher said once that pi is not an exact value unless you know all of its decimals, and as there are infinite number of decimals, pi cannot be exact by itself. He showed me himself with a calculator: HE entered pi with as many decimals that the window could hold. Then he subtracted this with the calculators internal constant of pi and Voilà, the result was not zero. "You see? pi is not an exact value!"

I wouldn't agree with him. You cannot represent pi exactly with decimals, but that's a property of the system you're using to represent it, not of pi itself. You could imagine a number system based on pi where it would be easy to represent pi exactly (but impossible to represent what we call "one" ).

This is all getting a bit abstract though 🙂

Originally posted by FabianFnas
Wait a little. Do I hear someone saying that you can use a calculator to prove any identity (like 2+2=4) if the expressions are in constants and not variables?

A calculator can't even prove that (1/3)*3=1!
A calculator cannot hold the exact value of 1/3 in its memory, only a approximation. Therefore, multiplying this approximate number with 3 ...[text shortened]... all. We can use a calculator to make it probable that it is an identity, but not as a proof.

Calculators can indeed hold fractions in memory, as fractions. My casiofx-300ES even calculates in fractions so it can find exact solutions. My HP48 does the same. If you command it to output in fractions it will so you can have an output of 1/3 which is exact.

Originally posted by FabianFnas
I think you have to put your calculator in some sort of fractional mode in order to calculate fractions exactly. Am I right in this?

You can't get a fraction when operating decimals but you can enter a fraction and it gets stored as a fraction, as two numbers.

Originally posted by mtthw:
"True, but an application such as MathCad that can do symbolic manipulation can handle that with no difficulties."

...but it doesn't store the numbers numerically, only symbolically. And that's one way of storing pi exactly, by storing it with a symbol representing pi exactly, not storing pi itself.
(What do I really know, I don't know anything how symbolic expressions are stored other than by a literal of the expression itself.)

Originally posted by mtthw:
"I wouldn't agree with him. You cannot represent pi exactly with decimals, ..."

He didn't know what he was talking about. He just took a chance, in order to not losing his face, hoping I wouldn't catch the wild guess.

Originally posted by sonhouse:
"Calculators can indeed hold fractions in memory, as fractions."

Originally posted by kbaumen:
"You can't get a fraction when operating decimals but you can enter a fraction and it gets stored as a fraction, as two numbers."

So it can hold 1/pi exactly? As a fraction?
What happens if you multiply it by pi? Do you get exactly =1 or is it rounding it to =1 only? Subtract 1 and see the result. If you get the result other than zero then we know that it rounds. Else we don't know anything. (?)

I think fraction means some sort of p/q where p and q are integers and q not zero. Therefore 1/pi is not a fraction in the mind of a calculator.

Originally posted by FabianFnas
I think fraction means some sort of p/q where p and q are integers and q not zero. Therefore 1/pi is not a fraction in the mind of a calculator.

That's how it goes with my calculator.

Besides the calculator stuff, can you prove the identity?

Originally posted by FabianFnas
Originally posted by mtthw:
[b]"True, but an application such as MathCad that can do symbolic manipulation can handle that with no difficulties."

...but it doesn't store the numbers numerically, only symbolically. And that's one way of storing pi exactly, by storing it with a symbol representing pi exactly, not storing pi itself.
(What do ...[text shortened]... rs and q not zero. Therefore 1/pi is not a fraction in the mind of a calculator.[/b]

In my calculator, pi is stored as pi, exact value. It isn' t stored as a decimal, but just a pi, a constant When i type in 1/pi, it stores it as 1/pi, simple as that, no decimals invovled at all. When i multiply by pi, it knows to cancel the pi, giving me a 1, exact value. If my calculator has to calculate e^pi-pi, it would not give me 19.999, it would give me e^pi-pi, unless i specifically ask it to give a decimal answer.

Originally posted by Dejection
In my calculator, pi is stored as pi, exact value. It isn' t stored as a decimal, but just a pi, a constant When i type in 1/pi, it stores it as 1/pi, simple as that, no decimals invovled at all. When i multiply by pi, it knows to cancel the pi, giving me a 1, exact value. If my calculator has to calculate e^pi-pi, it would not give me 19.999, it would give me e^pi-pi, unless i specifically ask it to give a decimal answer.

Can it hold any real number exactly?
What kind of calculator do you have?