# How do you find an expression for #sin(x)# in terms of #e^(ix)# and #e^(ix)#?

##### 3 Answers

#### Explanation:

Start from the MacLaurin series of the exponential function:

so:

Separate now the terms for

Note now that:

so:

and we can recognize the MacLaurin expansions of

which is Euler's formula.

Considering that

then:

and finally:

Other approach to problem. See below

#### Explanation:

We know that

Similarly,

But we know that

Then we have

Adding both identities

Subtarcting both, we have

Compare the Maclaurin series of

#### Explanation:

I assume the final formula in the question should read

Compare the Maclaurin series of

http://blogs.ubc.ca/infiniteseriesmodule/units/unit-3-power-series/taylor-series/maclaurin-expansion-of-sinx/

http://www.songho.ca/math/taylor/taylor_exp.html

Note that both of these series are convergent over the whole range of

We can immediately see that the terms in the sine series are very similar to those in the exponential series - they're the same size where they exist, but often have the opposite sign, and half of them are missing.

Recalling that the powers of

which becomes

To remove every second term, we combine it with the series for

which becomes (be careful combining minus signs and

When we take the difference of these series term by term, we get closer to what we want (NB taking the sum of them gives us a relation for

which is just the series above for

Compare at this point the hyperbolic functions, which you may have been introduced to already. In particular, note the definition of

The hyperbolic functions are a set of functions closely related to the trig functions via these formulae. As you progress with differential equations, you'll encounter situations where a simple change of sign to a coefficient makes the difference between finding trig function and hyperbolic function solutions. The relation between the two sets of functions is an important one.