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Aliun [14]
3 years ago
14

Find the polar form of 1+ i / 1- i ​

Mathematics
1 answer:
pickupchik [31]3 years ago
8 0

hmmm let's start off by using the conjugate of the denominator to <u>rationalize the denominator</u>.

\bf \cfrac{1+i}{1-i}\implies \cfrac{1+i}{1-i}\cdot \cfrac{1+i}{1+i}\implies \cfrac{(1+i)^2}{\underset{\textit{difference of squares}}{(1-i)(1+i)}}\implies \cfrac{\stackrel{FOIL}{1^2+2i+i^2}}{1^2-i^2} \\\\\\ \stackrel{\textit{recalling that }i^2=-1}{\cfrac{1+2i+(-1)}{1-(-1)}}\implies \cfrac{2i}{1+1}\implies \cfrac{~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~i}{~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\implies i\implies \stackrel{\textit{a+bi form}}{0 + 1i}

so the equation is really "i" in disguise, now, our coordinates are 0 and 1 for that complex point, we can't quite use tan⁻¹ to get the angle, since denominator is 0, meaning the point is lying on one of the axis, well, the rectangular (0,1) is right up above on the y-axis, namely at π/2.

\bf \begin{cases} r = \sqrt{0^2+1^2}\\ \qquad 1\\ \theta =\frac{\pi }{2} \end{cases} \qquad \qquad 1\left[cos\left( \frac{\pi }{2} \right) +i~sin\left( \frac{\pi }{2} \right) \right]

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Answer:

Step-by-step explanation:

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4 0
3 years ago
The ratio of my action to comedy movies is 4:5. If I have 35 comedy movies, how many are action? *
iogann1982 [59]

Answer:

28 action movies

Step-by-step explanation:

  1. Set up a proportion: \frac{4}{5} = \frac{x}{35}  
  2. Cross multiply, then divide: 35 × 4 = 140, 140 ÷ 5 = 28
  3. So, if you have 35 comedy movies, you would have 28 action movies.

I hope this helps!

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3 years ago
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GenaCL600 [577]

9514 1404 393

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  yes, no, no

Step-by-step explanation:

The triangle will be a right triangle if the sum of the squares of the two short sides, less the square of the long side, is zero. (This is what the Pythagorean theorem tells you.) The angle where the radius meets the circle must be a right angle if the line is to be a tangent.

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a) 3^2 +4^2 -5^2 = 9 + 16 -25 = 0 . . . . AB is a tangent

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b) 9^2 +15^2 -18^2 = 81 +225 -324 = -18 . . . . AB is not a tangent

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3 years ago
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Answer:

The answers in order are:

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These are the definitions of the terms.

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3 years ago
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