Can someone please show me step by step how to calculate the modulus and the argument for these complex numbers? I have a trig final next week and I am trying to finish this study guide so I am prepared. Thank you so much! If you can, please show my how you get the answer so I actually learn something. :)
a) -5 + 5i
b) -5
c)-3 -4i
think back to your polar coordinates. Same thing.
(a+bi) = r cisθ
where
tanθ = y/x
r^2 = a^2 + b^2
-5+5i
r^2 = 5^2+5^2, so r=5√2
tanθ = -5/5 = -1
Now, cosθ=x/r sinθ=y/r
since x<0 and y>0, θ is in II, so
θ = 3π/4
-5 = -5+0i = 5 cis π
-3-4i = 5 cis π+arctan(-4/-3)
because we are in QIII
Of course! I'd be happy to help you learn. To calculate the modulus and argument of a complex number, we can use the Cartesian form of complex numbers, which is written as a + bi, where "a" represents the real part and "b" represents the imaginary part.
a) Let's start with complex number -5 + 5i.
Step 1: Calculate the modulus.
The modulus, also known as the absolute value or magnitude of a complex number, represents the distance of the complex number from the origin (0,0) on the complex plane. To calculate the modulus, we can use the Pythagorean theorem.
Modulus = sqrt(a^2 + b^2)
For -5 + 5i, the modulus would be:
Modulus = sqrt((-5)^2 + (5)^2)
= sqrt(25 + 25)
= sqrt(50)
= 5sqrt(2)
So the modulus of -5 + 5i is 5sqrt(2).
Step 2: Calculate the argument.
The argument of a complex number, also known as the angle, represents the direction of the complex number in the complex plane. To calculate the argument, we use trigonometry.
Argument = arctan(b/a)
For -5 + 5i, the argument would be:
Argument = arctan(5/(-5))
= arctan(-1)
= -π/4 (in radians)
So the argument of -5 + 5i is -π/4 (in radians).
b) Now let's consider the complex number -5, which is a purely real number with no imaginary part.
Step 1: Calculate the modulus.
The modulus of a real number is simply the absolute value of the real number itself.
Modulus = |a|
For -5, the modulus would be:
Modulus = |-5|
= 5
So the modulus of -5 is 5.
Step 2: Calculate the argument.
The argument of a purely real number is either 0 or π, depending on whether the number is positive or negative.
Argument = 0 (if a > 0) or Argument = π (if a < 0)
Since -5 is negative, the argument of -5 is π.
c) Lastly, let's find the modulus and argument of -3 - 4i.
Step 1: Calculate the modulus.
Modulus = sqrt(a^2 + b^2)
For -3 - 4i, the modulus would be:
Modulus = sqrt((-3)^2 + (-4)^2)
= sqrt(9 + 16)
= sqrt(25)
= 5
So the modulus of -3 - 4i is 5.
Step 2: Calculate the argument.
Argument = arctan(b/a)
For -3 - 4i, the argument would be:
Argument = arctan((-4)/(-3))
= arctan(4/3)
To find the exact value of arctan(4/3), we can use a calculator or refer to a table of common trigonometric values.
Approximately, the argument of -3 - 4i is 0.93 radians (rounded to two decimal places).
I hope this explanation helps you understand how to calculate the modulus and argument of complex numbers. Good luck with your trigonometry final!