3. find the four angles that define the fourth root of z1=1+ sqrt3*i
z = 2 * (1/2 + i * sqrt(3)/2)
z = 2 * (cos(pi/3 + 2pi * k) + i * sin(pi/3 + 2pi * k))
z = 2 * (cos((pi/3) * (1 + 6k)) + i * sin((pi/3) * (1 + 6k)))
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6k)) + i * sin((pi/12) * (1 + 6k)))
pi/12, 7pi/12, 13pi/12, 19pi/2
4. what are the fourth roots of z1=sqrt3 +i
2^(1/4) * (cos(pi/12) + i * sin(pi/12))
2^(1/4) * (cos(7pi/12) + i * sin(7pi/12))
2^(1/4) * (cos(13pi/12) + i * sin(13pi/12))
2^(1/4) * (cos(19pi/12) + i * sin(19pi/12))
Hmmm. YOu did this the hard way, very hard way.
Conver to polar..
2 * (1/2 + i * sqrt(3)/2) =2*sqrt(1/4+3/4)= 2*sqrt1=2
angle: arctan sqrt3/(2*1/2)=arctansqrt3= 1.04 radians= PI/3
so the vector is 2@PI/3
the fourth root is 2^1/4 @PI/12
and 2^1/4 at (Pi/12 +Pi/2) or angle (PI/12+PI) or (PI/12 + 3PI/4)
that's how they taught us. Is it right?
To find the four angles that define the fourth root of z1=1+ sqrt3*i, we can use the formula:
z = r * (cos(theta + 2n*pi) + i * sin(theta + 2n*pi))
where z = 1+sqrt(3)*i, r = sqrt(1^2 + sqrt(3)^2) = 2, and theta = arctan(sqrt(3)/1) = pi/3.
1. For n = 0:
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6*0)) + i * sin((pi/12) * (1 + 6*0)))
= 2^(1/4) * (cos(pi/12) + i * sin(pi/12))
= pi/12
2. For n = 1:
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6*1)) + i * sin((pi/12) * (1 + 6*1)))
= 2^(1/4) * (cos(7pi/12) + i * sin(7pi/12))
= 7pi/12
3. For n = 2:
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6*2)) + i * sin((pi/12) * (1 + 6*2)))
= 2^(1/4) * (cos(13pi/12) + i * sin(13pi/12))
= 13pi/12
4. For n = 3:
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6*3)) + i * sin((pi/12) * (1 + 6*3)))
= 2^(1/4) * (cos(19pi/12) + i * sin(19pi/12))
= 19pi/12
Therefore, the four angles that define the fourth root of z1=1+sqrt(3)*i are pi/12, 7pi/12, 13pi/12, and 19pi/12.
For the fourth roots of z1=sqrt(3)+i, we can use the same formula with z = sqrt(3)+i.
1. For n = 0:
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6*0)) + i * sin((pi/12) * (1 + 6*0)))
= 2^(1/4) * (cos(pi/12) + i * sin(pi/12))
= pi/12
2. For n = 1:
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6*1)) + i * sin((pi/12) * (1 + 6*1)))
= 2^(1/4) * (cos(7pi/12) + i * sin(7pi/12))
= 7pi/12
3. For n = 2:
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6*2)) + i * sin((pi/12) * (1 + 6*2)))
= 2^(1/4) * (cos(13pi/12) + i * sin(13pi/12))
= 13pi/12
4. For n = 3:
z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6*3)) + i * sin((pi/12) * (1 + 6*3)))
= 2^(1/4) * (cos(19pi/12) + i * sin(19pi/12))
= 19pi/12
Therefore, the fourth roots of z1=sqrt(3)+i are pi/12, 7pi/12, 13pi/12, and 19pi/12.
To find the four angles that define the fourth roots of a complex number, you can follow these steps:
1. Express the complex number in the form a + bi.
For example, z1 = 1 + √3i.
2. Convert the complex number into polar form.
z1 = r(cos θ + i sin θ), where r is the magnitude and θ is the argument.
In this case, the magnitude r = √(a^2 + b^2) and the argument θ = tan^(-1)(b/a).
In our example, z1 = 2 * (cos(π/3) + i sin(π/3)).
3. Express the complex number in exponential form.
z1 = r * e^(iθ), where e is Euler's number and i is the imaginary unit.
In our example, z1 = 2 * e^(iπ/3).
4. Write the fourth root formula for complex numbers in exponential form.
The formula is z^(1/4) = r^(1/4) * e^(iθ/4).
5. Calculate the magnitude of the fourth root.
In our example, the magnitude of z^(1/4) is 2^(1/4) = √(2).
6. Calculate the argument of the fourth root.
Divide the original argument by 4 to obtain four different values.
In our example, the four arguments are (π/3)/4, (π/3 + 2π)/4, (π/3 + 4π)/4, and (π/3 + 6π)/4.
Simplifying these expressions gives you π/12, 7π/12, 13π/12, and 19π/12.
So, the four angles that define the fourth root of z1=1+√3i are π/12, 7π/12, 13π/12, and 19π/12.
To find the fourth roots of z1 = √3 + i, use the same steps:
1. Express the complex number in the form a + bi.
In this case, z1 = √3 + i.
2. Convert the complex number into polar form.
To find the magnitude r, use the formula r = √(a^2 + b^2), which gives r = √(3 + 1) = 2.
To find the argument θ, use the formula θ = tan^(-1)(b/a), which gives θ = tan^(-1)(1/√3) = π/6.
Therefore, z1 = 2 * (cos(π/6) + i sin(π/6)).
3. Express the complex number in exponential form.
z1 = 2 * e^(iπ/6).
4. Apply the fourth root formula for complex numbers in exponential form.
z1^(1/4) = 2^(1/4) * e^(i(π/6)/4) = 2^(1/4) * e^(iπ/24).
5. Calculate the magnitude of the fourth root.
In this case, the magnitude of z1^(1/4) is 2^(1/4) = √(2).
6. Calculate the argument of the fourth root.
Divide the original argument by 4 to obtain four different values.
In this case, the four arguments are (π/6)/4, (π/6 + 2π)/4, (π/6 + 4π)/4, and (π/6 + 6π)/4.
Simplifying these expressions gives you π/24, 7π/24, 13π/24, and 19π/24.
Therefore, the fourth roots of z1=√3 + i are 2^(1/4) * (cos(π/24) + i sin(π/24)), 2^(1/4) * (cos(7π/24) + i sin(7π/24)), 2^(1/4) * (cos(13π/24) + i sin(13π/24)), and 2^(1/4) * (cos(19π/24) + i sin(19π/24)).