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

Let us begin with the exponential function's MacLaurin series:

so:

Now take note of this:

so:

This is the formula for Euler.

then:

and lastly:

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Other approach to problem. See below

Next, we have

Including both personas

Combining the two, we have

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Compare the Maclaurin series of

It is evident right away that the terms in the sine series and the exponential series are nearly identical in size when they occur, but frequently have the opposite sign and have half of their terms missing.

As you work with differential equations, you'll come across scenarios where a small sign change in a coefficient determines whether trig function and hyperbolic function solutions can be found. This relationship between the two sets of functions is crucial. The hyperbolic functions are a collection of functions that are closely related to the trig functions through these formulae.

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[ \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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