Find the complex number [math: z] that satisfies [math: z5=−3+8i]z^5=-3+8i and [math: 2π5

Question

Find the complex number [math: z] that satisfies [math: z5=−3+8i]z^5=-3+8i and [math: 2π5<arg⁡z<4π5]\dfrac {2\pi }{5}<\arg {z}<\dfrac {4\pi }{5}. (Correct the answer to 2 decimal places.)

BlackTom AI · Accepted Answer

We are given z^5 = -3 + 8i and the argument constraint (2π/5) < arg z < (4π/5). To solve, first find the modulus and argument of -3 + 8i.
- Modulus: | -3 + 8i | = sqrt((-3)^2 + 8^2) = sqrt(9 + 64) = sqrt(73) ≈ 8.544.
- Principal argument: arg(-3 + 8i) lies in the second quadrant. Compute the reference angle arctan(|8/(-3)|) = arctan(8/3) ≈ 1.212. The principal argument is π − 1.212 ≈ 1.9296 rad.

If z^5 = re^{iθ} with r = sqrt(73) and θ ≈ 1.9296, then the fifth roots have modulus ρ = r^{1/5} and arguments φ_k = (θ + 2πk)/5 for k = 0,1,2,3,4.
- Compute ρ: ρ = (sqrt(73))^{1/5} ≈ 8.544^{0.2} ≈ 1.54.
- Compute the five candidate arguments:
  φ_0 = (1.9296)/5 ≈ 0.3859 rad
  φ_1 = (1.9296 + 2π)/5 ≈ (1.9296 + 6.2832)/5 ≈ 1.6425 rad
  φ_2 = (1.9296 + 4π)/5 ≈ (1.9296 + 12.5664)/5 ≈ 2.8991 rad
  φ_3 = (1.9296 + 6π)/5 ≈ (1.9296 + 18.8496)/5 ≈ 4.1558 rad
  φ_4 = (1.9296 + 8π)/5 ≈ (1.9296 + 25.1327)/5 ≈ 5.4125 rad

Now check the given constraint  (2π/5) < arg z < (4π/5). Numerically, 2π/5 ≈ 1.2566 rad and 4π/5 ≈ 2.5133 rad. Among φ_k, only φ_1 ≈ 1.6425 rad lies in the interval (1.2566, 2.5133).

Therefore, the solution is the root with k = 1:
- z = ρ [cos(φ_1) + i sin(φ_1)] ≈ 1.54 [cos(1.6425) + i sin(1.6425)].
- Numerically, cos(1.6425) ≈ -0.071 and sin(1.6425) ≈ 0.997, so z ≈ -0.11 + 1.54i.
- In polar form with two decimals, z ≈ 1.54 (cos 1.64 + i sin 1.64).

Now, evaluate the provided answer options (if any):
- The option 1.54(cos 2.68 + i sin 2.68) corresponds to a root with magnitude 1.54 but an angle of about 2.68 rad, which does not match φ_1 = 1.6425 rad and does not lie in the specified arg range. Therefore this option is not correct.
- The option -1.38 + 0.89i has magnitude sqrt( (-1.38)^2 + (0.89)^2 ) ≈ sqrt(1.9044 + 0.7921) ≈ sqrt(2.6965) ≈ 1.64, and its argument is in the second quadrant but its angle does not correspond to any φ_k = (1.9296 + 2πk)/5. Thus this option is also incorrect.

In summary, the correct root in the specified angular interval is z ≈ 1.54 [cos(1.64) + i sin(1.64)] ≈ -0.11 + 1.54i, but the given choices do not precisely match this root.

Find the complex number [math: z] that satisfies [math: z5=−3+8i]z^5=-3+8i and [math: 2π5<arg⁡z<4π5]\dfrac {2\pi }{5}<\arg {z}<\dfrac {4\pi }{5}. (Correct the answer to 2 decimal places.)简答题

类似问题

Given that − 2 + 2 𝑖 is a root, which of the follwing options lists all roots of the following polynomial: 𝑃 ( 𝑥 ) = 𝑧 3 + 8 𝑧 2 + 24 𝑧 + 32.

Which of the following is a 4th-roots of \(z = 1-\sqrt{3}i\)?

How did Egyptian culture and governance change during the Ptolemaic period compared to the pharaonic period?

Name the Supreme Court decision that removed financial constraints on PACs.

Political scientists Joshua Kalla and David Broockman find that senior policy makers made themselves available to _____ between three and four times more often than they did to constituents.

Name the political scientist who argues that parties are designed to serve the needs of ambitious politicians (namely candidates and officeholders).

Name the political scientist who said this famous quote about the problems of pluralism: “The flaw in the pluralist heaven is that the heavenly choir sings with a strong upper-class accent.”

What is the name of the third party candidate in 2000 who was not allowed to participate in the presidential debates? Hint: We watched a video clip with him in class.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单