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HACTEHA [7]
3 years ago
7

Find the complex fourth roots of 81(cos(3π/8)+isin(3π/8)). a) Find the fourth root of 81. b) Divide the angle in the problem by

4 to find the first argument. c)Use the fact that adding 2π to the angle 3π/8 produces the same effective angle to generate the other three possible angle for the fourth roots. d) Find all four of the fourth roots of 81(cos(3π/8)+isin(3π/8)). express your answer in polar form.
Mathematics
1 answer:
WARRIOR [948]3 years ago
3 0

Answer:

The answer is below

Step-by-step explanation:

Let a complex z = r(cos θ + isinθ), the nth root of the complex number is given as:

z_1=r^{\frac{1}{n} }(cos(\frac{\theta +2k\pi}{n} )+isin(\frac{\theta +2k\pi}{n} )),\\k=0,1,2,.\ .\ .,n-1

Given the complex number z = 81(cos(3π/8)+isin(3π/8)), the fourth root (i.e n = 4) is given as follows:

z_{k=0}=81^{\frac{1}{4} }(cos(\frac{\frac{3\pi}{8}  +2(0)\pi}{4} )+isin(\frac{\frac{3\pi}{8}  +2(0)\pi}{4} ))=3[cos(\frac{3\pi}{32} )+isin(\frac{3\pi}{32})] \\z_{k=0}=3[cos(\frac{3\pi}{32} )+isin(\frac{3\pi}{32})]\\\\z_{k=1}=81^{\frac{1}{4} }(cos(\frac{\frac{3\pi}{8}  +2(1)\pi}{4} )+isin(\frac{\frac{3\pi}{8}  +2(1)\pi}{4} ))=3[cos(\frac{19\pi}{32} )+isin(\frac{19\pi}{32})] \\z_{k=1}=3[cos(\frac{19\pi}{32} )+isin(\frac{19\pi}{32})]\\\\

z_{k=2}=81^{\frac{1}{4} }(cos(\frac{\frac{3\pi}{8}  +2(2)\pi}{4} )+isin(\frac{\frac{3\pi}{8}  +2(2)\pi}{4} ))=3[cos(\frac{35\pi}{32} )+isin(\frac{35\pi}{32})] \\z_{k=2}=3[cos(\frac{35\pi}{32} )+isin(\frac{35\pi}{32})]\\\\z_{k=3}=81^{\frac{1}{4} }(cos(\frac{\frac{3\pi}{8}  +2(3)\pi}{4} )+isin(\frac{\frac{3\pi}{8}  +2(3)\pi}{4} ))=3[cos(\frac{51\pi}{32} )+isin(\frac{51\pi}{32})] \\z_{k=3}=3[cos(\frac{51\pi}{32} )+isin(\frac{51\pi}{32})]

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Please give me the correct answer ​
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Answer:

PQ = 5 units

QR = 8 units

Step-by-step explanation:

Given

P(-3, 3)

Q(2, 3)

R(2, -5)

To determine

The length of the segment PQ

The length of the segment QR

Determining the length of the segment PQ

From the figure, it is clear that P(-3, 3) and Q(2, 3) lies on a horizontal line. So, all we need is to count the horizontal units between them to determine the length of the segments P and Q.

so

P(-3, 3), Q(2, 3)

PQ = 2 - (-3)

PQ = 2+3

PQ = 5 units

Therefore, the length of the segment PQ = 5 units

Determining the length of the segment QR

Q(2, 3), R(2, -5)

(x₁, y₁) = (2, 3)

(x₂, y₂) = (2, -5)

The length between the segment QR is:

l=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}

  =\sqrt{\left(2-2\right)^2+\left(-5-3\right)^2}

  =\sqrt{0+8^2}

  =\sqrt{8^2}

Apply radical rule: \sqrt[n]{a^n}=a,\:\quad \mathrm{\:assuming\:}a\ge 0

  =8

Therefore, the length between the segment QR is: 8 units

Summary:

PQ = 5 units

QR = 8 units

8 0
2 years ago
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