Blog: Can you help me understand how to find the trigonometric form of -4?

Sunday, March 17, 2013

Can you help me understand how to find the trigonometric form of -4?

Hello!

I'll try.

The trigonometric form of a complex number is

$\displaystyle{r}{\left({\cos{{\left(\phi\right)}}}+{i}\cdot{\sin{{\left(\phi\right)}}}\right)},$

where $\displaystyle{r}$ is a positive number and $\displaystyle\phi$ is a number in $\displaystyle{\left[{0},{2}\pi\right)}.$

Here $\displaystyle{r}$ is the absolute value of a complex number and $\displaystyle\phi$ is the argument. Any nonzero complex number has one and only one such representation.

If we plot a complex number on a coordinate plane and draw a directed segment from the origin to the number's point,...

Hello!

I'll try.

The trigonometric form of a complex number is

$\displaystyle{r}{\left({\cos{{\left(\phi\right)}}}+{i}\cdot{\sin{{\left(\phi\right)}}}\right)},$

where $\displaystyle{r}$ is a positive number and $\displaystyle\phi$ is a number in $\displaystyle{\left[{0},{2}\pi\right)}.$

Here $\displaystyle{r}$ is the absolute value of a complex number and $\displaystyle\phi$ is the argument. Any nonzero complex number has one and only one such representation.

If we plot a complex number on a coordinate plane and draw a directed segment from the origin to the number's point, then $\displaystyle{r}$ will be the length of this segment and $\displaystyle\phi$ will be the angle between the positive half of the x-axis and the segment.

For the number $\displaystyle-{4}=-{4}+{0}{i}$ the absolute value is obviously $\displaystyle{4}=\sqrt{{{{\left(-{4}\right)}}^{{2}}+{{0}}^{{2}}}}.$

What about $\displaystyle\phi$ ? The point $\displaystyle{\left(-{4},{0}\right)}$ lies at the negative half of the x-axis and the angle from the positive half to the negative is obviously $\displaystyle\pi$ (a half of an entire circumference, which is $\displaystyle{2}\pi$ ).

We can check this: $\displaystyle{4}{\left({\cos{{\left(\pi\right)}}}+{i}\cdot{\sin{{\left(\pi\right)}}}\right)}={4}\cdot{\left(-{1}+{0}\cdot{i}\right)}=-{4}$ (true).

So the answer is $\displaystyle-{4}={4}{\left({\cos{{\left(\pi\right)}}}+{i}\cdot{\sin{{\left(\pi\right)}}}\right)}.$

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Sunday, March 17, 2013

Can you help me understand how to find the trigonometric form of -4?

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What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?

Sunday, March 17, 2013

Can you help me understand how to find the trigonometric form of -4?

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Post a Comment

What is the Exposition, Rising Action, Climax, and Falling Action of &quot;One Thousand Dollars&quot;?

What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?