Suppose that z1=6-8i. Find:
A. The Trig Form of the complex number z1, where your theta is in degrees.
B. The Trig form of z1*z2, where
z2=5[cos(60degrees)+isin(60degrees)]
C. The Trig Form of (z1)^4
looks like z1 is in QIV, since x>0 and y<0
so, θ = 360 - arctan(8/6) = 360 - 53.1 = 306.9°
z1 = 10 cis 306.9°
z2 = 5 cis 60°
z1*z2 = 10*5 cis (-53.1+60) = 50 cis 6.9°
z1^4 = 10^4 cis 4*(-53.1)
= 10000 cis -212.4°
= 10000 cis 147.6°
x = 10000 cos 147.6° = -8443
y = 10000 sin 147.6° = 5358
z1^4 = -8443 + 5359i
To find the Trig Form of the complex number z1 = 6-8i (where theta is in degrees), we need to determine the magnitude (r) and the angle (theta) of z1.
A. Finding the Trig Form of z1:
Step 1: Calculate the magnitude (r):
The magnitude of a complex number can be calculated using the formula: r = sqrt(a^2 + b^2), where a is the real part and b is the imaginary part of the complex number.
In this case, a = 6 and b = -8.
r = sqrt((6)^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10.
Step 2: Calculate the angle (theta):
The angle of a complex number can be calculated using the formula: tan(theta) = b/a, where a is the real part and b is the imaginary part of the complex number.
In this case, a = 6 and b = -8.
tan(theta) = (-8)/6 = -4/3.
To determine the angle (theta), we can use the arctan function (or inverse tangent):
theta = arctan(-4/3) ≈ -53.13 degrees.
Therefore, the Trig Form of z1 = 6-8i is: z1 = 10(cos(-53.13 degrees) + isin(-53.13 degrees)).
Now let's move on to part B:
B. Finding the Trig Form of z1*z2:
Given z2 = 5(cos(60 degrees) + isin(60 degrees)), we can calculate the product z1*z2 as follows:
z1*z2 = (6-8i)*(5[cos(60 degrees)+isin(60 degrees)])
Step 1: Multiply the magnitudes:
The magnitudes of z1 and z2 are already provided: |z1| = 10 and |z2| = 5.
So, |z1*z2| = |z1| * |z2| = 10 * 5 = 50.
Step 2: Add the angles:
The angles (theta) of z1 and z2 are already provided: theta1 = -53.13 degrees and theta2 = 60 degrees.
So, theta = theta1 + theta2 = -53.13 + 60 = 6.87 degrees.
Therefore, the Trig Form of z1*z2 = 50(cos(6.87 degrees) + isin(6.87 degrees)).
Moving on to part C:
C. Finding the Trig Form of (z1)^4:
To find the Trig Form of (z1)^4, we need to take the given complex number z1 and raise it to the fourth power.
Step 1: Calculate the magnitude of z1^4:
The magnitude (r) of (z1)^4 will be (|z1|)^4.
So, r = (|z1|)^4 = (10)^4 = 10000.
Step 2: Calculate the angle of z1^4:
The angle (theta) of z1^4 will be 4 times the angle of z1.
So, theta = 4 * theta1 = 4 * (-53.13 degrees) = -212.5 degrees.
Therefore, the Trig Form of (z1)^4 = 10000(cos(-212.5 degrees) + isin(-212.5 degrees)).
Note: In all the trig forms provided, cos is used for the real part and sin is used for the imaginary part.