The definition on a rigorous level for sine and cosine is:

From here can you prove that,

If then,

And .

I do not know how to start this one.

Printable View

- April 4th 2006, 07:23 PMThePerfectHackerRigorous Sine and Cosine
The definition on a rigorous level for sine and cosine is:

From here can you prove that,

If then,

And .

I do not know how to start this one. - April 5th 2006, 09:46 AMTD!
I'd use the definitions which extend the arguments to the complex case.

- April 5th 2006, 01:16 PMThePerfectHackerQuote:

Originally Posted by**TD!**

then you can easily denomstarte that,

.

But what about the other two facts, the necessary and suffienct conditions for the zero's of the sine function?

I was thinking, show that there exists a non-zero number on some interval which is a zero of the sine function by the use of the indetermediate value theorem. Then, demonstate that is is a zero then so is but that does not prove the "only if" part. - April 5th 2006, 01:56 PMTD!
You can use that identity of course, but I was referring to the definition:

The exponential function is periodic, with period . We can now use this to find the zeroes.

Now we use the periodicy of e^z.

- April 5th 2006, 02:29 PMThePerfectHackerQuote:

Originally Posted by**TD!**

- April 6th 2006, 08:14 AMTD!
That is indeed easy to show by using sin & cos :o :D