# How do you simplify #3(cos((7pi)/3)+isin((7pi)/3))div(cos(pi/2)+isin(pi/2))# and express the result in rectangular form?

The answer is

We need Euler's relation

Therefore,

So,

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To simplify the expression (3(\cos\left(\frac{7\pi}{3}\right) + i\sin\left(\frac{7\pi}{3}\right)) \div (\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right))) and express the result in rectangular form, we'll use Euler's formula:

[ e^{i\theta} = \cos(\theta) + i\sin(\theta) ]

First, let's express both numerator and denominator in Euler's form:

Numerator: [ \cos\left(\frac{7\pi}{3}\right) + i\sin\left(\frac{7\pi}{3}\right) = e^{i\left(\frac{7\pi}{3}\right)} ]

Denominator: [ \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) = e^{i\left(\frac{\pi}{2}\right)} ]

Now, let's rewrite the expression using these forms:

[ \frac{3e^{i\left(\frac{7\pi}{3}\right)}}{e^{i\left(\frac{\pi}{2}\right)}} ]

Using the properties of exponents, when dividing exponential terms with the same base, we subtract the exponents:

[ 3e^{i\left(\frac{7\pi}{3} - \frac{\pi}{2}\right)} ]

[ 3e^{i\left(\frac{14\pi}{6} - \frac{3\pi}{6}\right)} ]

[ 3e^{i\left(\frac{11\pi}{6}\right)} ]

Now, we need to convert this expression back to rectangular form:

Using Euler's formula in reverse: [ e^{i\theta} = \cos(\theta) + i\sin(\theta) ]

We find: [ \cos\left(\frac{11\pi}{6}\right) + i\sin\left(\frac{11\pi}{6}\right) ]

[ \frac{\sqrt{3}}{2} - \frac{1}{2}i ]

Thus, the simplified expression in rectangular form is ( \frac{3\sqrt{3}}{2} - \frac{3}{2}i ).

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