# How do you evaluate # e^( ( pi)/4 i) - e^( ( 7 pi)/4 i)# using trigonometric functions?

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To evaluate ( e^{(\frac{\pi}{4}i)} - e^{(\frac{7\pi}{4}i)} ) using trigonometric functions, we can express the complex exponential terms in terms of trigonometric functions using Euler's formula.

Euler's formula states that ( e^{ix} = \cos(x) + i \sin(x) ), where ( i ) is the imaginary unit.

So, ( e^{(\frac{\pi}{4}i)} ) can be expressed as ( \cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}) ), and ( e^{(\frac{7\pi}{4}i)} ) can be expressed as ( \cos(\frac{7\pi}{4}) + i \sin(\frac{7\pi}{4}) ).

Now, evaluate these expressions:

( \cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i )

( \cos(\frac{7\pi}{4}) + i \sin(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i )

Now, subtract the second from the first:

( (\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i) - (\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i )

( = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + (\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2})i )

( = 0 + \sqrt{2}i )

So, ( e^{(\frac{\pi}{4}i)} - e^{(\frac{7\pi}{4}i)} = \sqrt{2}i ).

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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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