1. Find the modulus and argument of the following complex numbers, and write them in trigonometric form:
a. 5 – 8i
Answer = Sqrt{89} ( cos (-1.01219701) + i sin ( -1.01219701))
b. –1 – i
answer = Sqrt2 (cos (5pi/4) + i sin (5pi/4))
c. –5 + 12i
answer = 13 (cos ( -4.31759786068493) + i sin ( -4.31759786068493))
d. 1 + sqrt{3}i
answer = 2 ( cos (pi/3) + i sin (pi/3))
all look good, but I'm curious why you had negative angles for (a) annd (c).
For (a) I could see it, because the angle is greater than -pi
For (c) I would have added 2pi to get an angle less than pi, since the angle is in QII.
I think he/she has a negative angle for a) by entering the data into a calculator and simply blindly accepting the answer the calculator gives.
Remember the calculator is programmed to give the closest angle to zero.
so tan^-1 (-8/5) = -1.012....
I would take 2π - 1.012....
To find the modulus and argument of a complex number and write it in trigonometric form, you can use the following steps:
Step 1: Find the modulus (magnitude) of the complex number.
The modulus of a complex number is given by the formula: modulus = sqrt(real^2 + imaginary^2)
Step 2: Find the argument (angle) of the complex number.
The argument of a complex number is given by the formula: argument = atan(imaginary/real)
Step 3: Write the complex number in trigonometric form.
The trigonometric form of a complex number is written as: modulus * (cos(argument) + i*sin(argument))
Now let's apply these steps to each complex number given:
a. 5 - 8i:
Step 1: modulus = sqrt(5^2 + (-8)^2) = sqrt(89)
Step 2: argument = atan(-8/5) = -1.01219701
Step 3: Trigonometric form = sqrt(89) * (cos(-1.01219701) + i*sin(-1.01219701))
b. -1 - i:
Step 1: modulus = sqrt((-1)^2 + (-1)^2) = sqrt(2)
Step 2: argument = atan(-1/(-1)) = pi + atan(1) = 5pi/4
Step 3: Trigonometric form = sqrt(2) * (cos(5pi/4) + i*sin(5pi/4))
c. -5 + 12i:
Step 1: modulus = sqrt((-5)^2 + 12^2) = 13
Step 2: argument = atan(12/(-5)) = -4.31759786068493
Step 3: Trigonometric form = 13 * (cos(-4.31759786068493) + i*sin(-4.31759786068493))
d. 1 + sqrt(3)i:
Step 1: modulus = sqrt(1^2 + (sqrt(3))^2) = 2
Step 2: argument = atan(sqrt(3)/1) = pi/3
Step 3: Trigonometric form = 2 * (cos(pi/3) + i*sin(pi/3))
And there you have it! The modulus and argument (in radians) of the given complex numbers, written in trigonometric form.