Let z=−93√−9i.
When z is graphed in the complex plane, what radian angle does it make with the positive x-axis?
What radian angles do z2, z3, and z4 make with the positive x-axis?
z2:
z3:
z4:
For what values of n does z^n lie on the positive x-axis?
For the purposes of this problem, report all angles as their smallest positive equivalent.
your complex number is garbled.
To find the radian angle that a complex number makes with the positive x-axis, we can convert the complex number from rectangular form (z = a + bi) to polar form (z = r * exp(iθ)), where r is the magnitude of the complex number and θ is the angle it makes with the positive x-axis.
Let's start by finding the polar form of the given complex number z.
Given: z = -93√(-9i)
Step 1: Find the magnitude (r)
The magnitude (r) of a complex number in rectangular form is given by the formula: r = √(a^2 + b^2).
In this case, a = -93√(-9) and b = 0 (since there is no imaginary part).
Calculating the magnitude:
r = √((-93√(-9))^2 + 0^2)
= √((93^2 * √(-9)^2)
= √(93^2 * 9)
= √(93^2) * √(9)
= 93 * 3
= 279
So, the magnitude (r) of z is 279.
Step 2: Find the angle (θ)
To determine the angle (θ) in polar form, we can use the formula: θ = arctan(b / a).
Since there is no imaginary part in this case (b = 0), the angle will be 0 radians.
Therefore, z can be written in polar form as z = 279 * exp(i0).
Now, we can answer the questions:
1. When z is graphed in the complex plane, it makes an angle of 0 radians with the positive x-axis.
2. To find the radian angles z^2, z^3, and z^4 make with the positive x-axis, we need to raise z to the respective powers.
z^2 = (279 * exp(i0))^2 = 279^2 * exp(i0 * 2) = 77841 * exp(i0) = 77841
z^3 = (279 * exp(i0))^3 = 279^3 * exp(i0 * 3) = 21647799 * exp(i0) = 21647799
z^4 = (279 * exp(i0))^4 = 279^4 * exp(i0 * 4) = 61222009041 * exp(i0) = 61222009041
The radian angles z^2, z^3, and z^4 make with the positive x-axis are 0 radians as well.
3. To find the values of n for which z^n lies on the positive x-axis, we need to determine when the angle of the polar form is a multiple of π radians (180 degrees).
The equation for the angle (θ) in terms of n is: θ = n * π.
Solving for n:
0 = n * π
n = 0
Therefore, z^n lies on the positive x-axis when n equals 0.
In summary:
- The radian angle that z makes with the positive x-axis is 0 radians.
- The radian angles that z^2, z^3, and z^4 make with the positive x-axis are also 0 radians.
- The values of n for which z^n lies on the positive x-axis is when n equals 0.