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To find: The roots of the eighth root of 1 and sketch the roots in the complex plane.
Answer to Problem 37E
The roots of eighth root of 1 are wk=11n[cos(0+2kπ8)+isin(0+2kπ8)]=coskπ8+isinkπ8 where k=0,1,2,…,7.
Explanation of Solution
Theorem used:
Roots of a
Let z=r(cosθ+isinθ) and n be a positive integer. Then z has the n distinct nth roots wk=r1n[cos(θ+2kπn)+isin(θ+2kπn)] where k=0,1,2,…,n−1.
Calculation:
Rewrite the complex number 1 in polar form.
The polar form of the complex number z=a+bi is z=r(cosθ+isinθ) where r=|z|=√a2+b2 and tanθ=ba.
Consider the complex number 1.
Obtain the argument of the complex number 1.
tanθ=01=0
Thus, the argument of argument of the complex number 1 is θ=tan−1(0)=0
Obtain the modulus of the complex number 1.
r=|1|=√12+(0)2=√1=1
Thus, the value of r=1.
Therefore, the polar form of the complex number 1 is 1=(cos0+isin0).
By the above theorem, the roots of eighth root of 1 are wk=11n[cos(0+2kπ8)+isin(0+2kπ8)]=coskπ8+isinkπ8 where k=0,1,2,…,7.
Use online calculator to sketch the roots in the complex plane as shown below in Figure 1.
From figure 1, it is observed that all roots of eighth root of 1 form a circle on complex plane.
Chapter I Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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