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Sagot :
Let's solve the problem of identifying which point on the complex plane represents the given complex number in polar form: [tex]\( 4 \sqrt{2}\left(\cos \left(\frac{5 \pi}{4}\right)+i \sin \left(\frac{5 \pi}{4}\right)\right) \)[/tex].
The complex number is given in polar form: [tex]\( r (\cos(\theta) + i \sin(\theta)) \)[/tex].
In this case:
- [tex]\( r = 4 \sqrt{2} \)[/tex]
- [tex]\( \theta = \frac{5\pi}{4} \)[/tex]
First, we will convert the given complex number from polar to rectangular form.
1. Calculate [tex]\( \cos\left(\frac{5\pi}{4}\right) \)[/tex] and [tex]\( \sin\left(\frac{5\pi}{4}\right) \)[/tex]:
The angle [tex]\( \frac{5\pi}{4} \)[/tex] lies in the third quadrant, where both cosine and sine are negative.
- [tex]\( \cos\left(\frac{5\ \pi}{4}\right) = -\frac{\sqrt{2}}{2} \)[/tex]
- [tex]\( \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)[/tex]
2. Formulate the real and imaginary parts:
[tex]\( x = r \cos(\theta) = 4 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} \)[/tex]
[tex]\[ x = 4 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} = 4 \cdot -1 = -4 \][/tex]
[tex]\( y = r \sin(\theta) = 4 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} \)[/tex]
[tex]\[ y = 4 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} = 4 \cdot -1 = -4 \][/tex]
The rectangular form of the given complex number is therefore [tex]\( -4 - 4i \)[/tex].
3. Determine the point on the complex plane:
We have calculated the rectangular form of the complex number to be [tex]\( -4 - 4i \)[/tex]. We now need to compare this point to the given points:
- Point [tex]\(A\)[/tex] should be at coordinates [tex]\((x_1, y_1)\)[/tex]
- Point [tex]\(B\)[/tex] should be at coordinates [tex]\((x_2, y_2)\)[/tex]
- Point [tex]\(C\)[/tex] should be at coordinates [tex]\((x_3, y_3)\)[/tex]
- Point [tex]\(D\)[/tex] should be at coordinates [tex]\((x_4, y_4)\)[/tex]
Assuming the given points [tex]\(A, B, C, D\)[/tex] on the complex plane were provided in the problem statement, we would compare each one to the calculated coordinate [tex]\((-4, -4)\)[/tex].
The point on the complex plane that represents [tex]\( 4 \sqrt{2}\left(\cos \left(\frac{5 \pi}{4}\right)+i \sin \left(\frac{5 \pi}{4}\right)\right) \)[/tex] is the one with coordinates [tex]\((-4, -4)\)[/tex]. Based on this, you can identify the specific point [tex]\(A, B, C,\)[/tex] or [tex]\(D\)[/tex] from the provided information.
The complex number is given in polar form: [tex]\( r (\cos(\theta) + i \sin(\theta)) \)[/tex].
In this case:
- [tex]\( r = 4 \sqrt{2} \)[/tex]
- [tex]\( \theta = \frac{5\pi}{4} \)[/tex]
First, we will convert the given complex number from polar to rectangular form.
1. Calculate [tex]\( \cos\left(\frac{5\pi}{4}\right) \)[/tex] and [tex]\( \sin\left(\frac{5\pi}{4}\right) \)[/tex]:
The angle [tex]\( \frac{5\pi}{4} \)[/tex] lies in the third quadrant, where both cosine and sine are negative.
- [tex]\( \cos\left(\frac{5\ \pi}{4}\right) = -\frac{\sqrt{2}}{2} \)[/tex]
- [tex]\( \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)[/tex]
2. Formulate the real and imaginary parts:
[tex]\( x = r \cos(\theta) = 4 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} \)[/tex]
[tex]\[ x = 4 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} = 4 \cdot -1 = -4 \][/tex]
[tex]\( y = r \sin(\theta) = 4 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} \)[/tex]
[tex]\[ y = 4 \sqrt{2} \cdot -\frac{\sqrt{2}}{2} = 4 \cdot -1 = -4 \][/tex]
The rectangular form of the given complex number is therefore [tex]\( -4 - 4i \)[/tex].
3. Determine the point on the complex plane:
We have calculated the rectangular form of the complex number to be [tex]\( -4 - 4i \)[/tex]. We now need to compare this point to the given points:
- Point [tex]\(A\)[/tex] should be at coordinates [tex]\((x_1, y_1)\)[/tex]
- Point [tex]\(B\)[/tex] should be at coordinates [tex]\((x_2, y_2)\)[/tex]
- Point [tex]\(C\)[/tex] should be at coordinates [tex]\((x_3, y_3)\)[/tex]
- Point [tex]\(D\)[/tex] should be at coordinates [tex]\((x_4, y_4)\)[/tex]
Assuming the given points [tex]\(A, B, C, D\)[/tex] on the complex plane were provided in the problem statement, we would compare each one to the calculated coordinate [tex]\((-4, -4)\)[/tex].
The point on the complex plane that represents [tex]\( 4 \sqrt{2}\left(\cos \left(\frac{5 \pi}{4}\right)+i \sin \left(\frac{5 \pi}{4}\right)\right) \)[/tex] is the one with coordinates [tex]\((-4, -4)\)[/tex]. Based on this, you can identify the specific point [tex]\(A, B, C,\)[/tex] or [tex]\(D\)[/tex] from the provided information.
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