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## Precalculus

### Course: Precalculus>Unit 3

Lesson 5: Modulus (absolute value) and argument (angle) of complex numbers

# Complex number absolute value & angle review

Review your knowledge of the complex number features: absolute value and angle. Convert between them and the rectangular representation of a number.
Absolute value of $a+bi$$\mid \phantom{\rule{-0.167em}{0ex}}\phantom{\rule{-0.167em}{0ex}}z\phantom{\rule{-0.167em}{0ex}}\mid =\sqrt{{a}^{2}+{b}^{2}}$
Angle of $a+bi$$\theta ={\mathrm{tan}}^{-1}\left(\frac{b}{a}\right)$
Rectangular form from absolute value $r$ and angle $\theta$$r\mathrm{cos}\left(\theta \right)+r\mathrm{sin}\left(\theta \right)\cdot i$

## What are the absolute value and angle of complex numbers?

We are used to writing complex numbers in their rectangular form, that gives their $\text{real}$ and $\text{imaginary}$ parts. For example, $3+4i$.
We can plot numbers in the complex plane according to their parts:
Considered graphically, there's another way to uniquely describe complex numbers — their $\text{absolute value}$ and $\text{angle}$:
The $\text{absolute value}$, or $\text{modulus}$, gives the distance of the number from the origin in the complex plane, while its $\text{angle}$, or $\text{argument}$, is the angle the number forms with the positive Real axis.
The absolute value of a complex number $z$ is written in the same way as the absolute value of a real number, $|z|$.
Want to learn more about the absolute value and angle of complex numbers? Check out this video.

## Practice set 1: Finding absolute value

To find the absolute value of a complex number, we take the square root of the sum of the squares of the parts (this is a direct result of the Pythagorean theorem):
$|a+bi|=\sqrt{{a}^{2}+{b}^{2}}$
For example, the absolute value of $3+4i$ is $\sqrt{{3}^{2}+{4}^{2}}=\sqrt{25}=5$.
Problem 1.1
$|3+7i|=$

Want to try more problems like this? Check out this exercise.

## Practice set 2: Finding angle

To find the angle of a complex number, we take the inverse tangent of the ratio of its parts:
$\theta ={\mathrm{tan}}^{-1}\left(\frac{b}{a}\right)$
This results from using trigonometry in the right triangle formed by the number and the Real axis.

### Example 1: Quadrant $\text{I}$‍

Let's find the angle of $3+4i$:
${\mathrm{tan}}^{-1}\left(\frac{4}{3}\right)\approx {53}^{\circ }$

### Example 2: Quadrant $\text{II}$‍

Let's find the angle of $-3+4i$. First, notice that $-3+4i$ is in Quadrant $\text{II}$.
${\mathrm{tan}}^{-1}\left(\frac{4}{-3}\right)\approx -{53}^{\circ }$
$-{53}^{\circ }$ is in Quadrant $\text{IV}$, not $\text{II}$. We must add ${180}^{\circ }$ to obtain the opposite angle:
$-{53}^{\circ }+{180}^{\circ }={127}^{\circ }$
Problem 2.1
$z=1+4i$
$\theta =$
${}^{\circ }$
Round your answer, if necessary, to the nearest tenth. Express $\theta$ between $-{180}^{\circ }$ and ${180}^{\circ }$.

Want to try more problems like this? Check out this exercise.

## Practice set 3: Rectangular form from absolute value and angle

To find the real and imaginary parts of a complex number from its absolute value and angle, we multiply the absolute value by the sine or cosine of the angle:
$\stackrel{a}{\stackrel{⏞}{r\mathrm{cos}\left(\theta \right)}}+\stackrel{b}{\stackrel{⏞}{r\mathrm{sin}\left(\theta \right)}}\cdot i$
This results from using trigonometry in the right triangle formed by the number and the Real axis.
For example, this is the rectangular form of the complex number whose absolute value is $2$ and angle is ${30}^{\circ }$:
$2\mathrm{cos}\left({30}^{\circ }\right)+2\mathrm{sin}\left({30}^{\circ }\right)i=\sqrt{3}+1i$
Problem 3.1
$|{z}_{1}|=3$ and ${\theta }_{1}={20}^{\circ }$
${z}_{1}=$
+
$i$

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• I found a really easy way to solve the problems in the earlier exercise!
Take the cos of the angle and multiply it by the magnitude to get the x value (rounding it to the nearest thousandth) and use sin for the y value but do the same thing.
Example:
A complex number z₁ has a magnitude |z₁| = 20 and angle θ₁ = 281°
Express z₁ in rectangular form, as z₁ = a + bi.
Round a and b to the nearest thousandth.

a = cos(θ₁) × |z₁|
b = sin(θ₁) × |z₁|

My question is, is there anything wrong with the way that I solve this? Is there a problem that I could run into, using this?
• That looks correct. You have found a theorem - can you prove it?
Hint: try drawing it on a standard Cartesian plane and using trigonometry for the proof. Represent your complex number as a line starting at the origin, at an angle of theta, and with length |z|.
• Is there a way to know when to add or substract 180degrees or pi? For example, if the a is negative and the b is positive, is there a way to know exactly whether or not to add or subtract? And this for three other cases (both positive, both negative, a positive and b negative)?
• I believe I have found the answer to when to add/ subtract 180degrees/pi. I have tested my theory with numerous examples and it has proven to work:
If your point lands in Q1= no adding or subtracting, Q2= add 180 or pi, Q3= subtract 180 or pi, and Q4= no adding or subtracting.

• Does anyone know how to tell which quadrant the question was asking? e.g. Express
θ between −180 and 180 degree. I finished the trig, most went well but I'm always confused with the range they set up and always get the wrong answer.
This is how I understood the phrase "Express theta between -180 and 180 degree": I draw a unit circle, then go from 0 degree (which is the right side of the x-axis) to 180 degree (the left side of x-axis); then I do the same from 0 degree to -180 degree (so it is a clockwise rotation of 180 degree to the left side of x-axis). As the result I don't know which area/quadrants the question was asking because both 180 and -180 degree are on the same line (the left side of the x-axis).
• The 180 and -180 degree clarifications are there to ensure that you don't input wild values like 7928 degrees--stay within the boundaries, and you're fine.

As for the quadrants, it helps me to sketch the point. However, if you feel like memorizing, remember that a positive a and a negative b will always be in the fourth quadrant, and you can figure out the other three :)
• In some of the questions on the complex numbers from absolute value and angle practice, it instructs to list answers in exact terms. How is this obtained from the rcos(θ) + rsin(θ) * i formula?
• Some values of sine and cosine have values that you can figure out by hand such as trigonometric values in a 30-60-90 triangle or 45-45-90 triangles and other trig values you can derive from these triangles.
(1 vote)
• when i put in 2cos(30) into my calculator i get 1.732050808 and it was telling me thats wrong. It took me a while to figure out that if i square that number i get the square root of three. and thats the format that it accepts it as. i guess my question is that that what everyone else is doing is squaring there answere to see if it comes to a a natural number to determine what format to put it in?
• Can someone explain how to find inverse of trigo functions
• You have to use a calculator for this particular kind of operation, usually labeled as arc then trig function (e.g. arccos(x)) or the trig function raised to the negative one power (e.g. sin^-1(x)).
• hi, i want to ask one question. How to solve this type of question?
Question:
The points P and Q in an Argand diagram represent the complex number 8-i and 12+6i respectively and O is the origin. Show that the triangle OPQ is isosceles and calculate the size of angle OPQ correct to the nearest degree.
• First we find all side lengths of the triangle. Let 𝑝 = 8 – 𝑖 and 𝑞 = 12 + 6𝑖. Then:
𝑃𝑂 = |𝑝| = √65
𝑄𝑂 = |𝑞| = 6√5
𝑃𝑄 = |𝑝 – 𝑞| = √65
Thus ∆𝑃𝑄𝑂 is isosceles.
Finding ∠𝑂𝑃𝑄 is easy with the Law of Cosines:
180 = 65 + 65 – 2 • 65²cos 𝑃
This gives:
cos 𝑃 = -25/65²
Or:
cos 𝑃 = -1/169
Taking the arccosine of both sides gives 𝑃 ≈ 90.34˚ which is simply 90˚ when rounded to the nearest degree.
Comment if you have questions!
• Why do we need to add 180 degrees in Example 2: Quadrant ||
• Think of it this way: the angle is how far the modulus/absolute value is from the positive side of the real number axis. When the modulus is in Quadrant 1, like in the example before Example 2, it is only 53 degrees away from it. Whereas in Example 2, the modulus is an additional 180 degrees to the left because it is in Quadrant 2, which is why we need to add 180 degrees. Does that make sense?
• how do we know when to subtract 180 degrees from theta and when i should add 180 degrees to theta? Please help!
• If the angle theta lies in quadrant 2 you must add 180 and quadrant 3 substract 180
• Usually I do fine in math, but finding the rectangular form in exact terms (as in the form a+b i where a = 5 times the square root of 3 and b = the square root of 7, for example) is really stumping me. So far, I have only got one problem right, and that was through trial and error. How exactly am I supposed to find the rectangular form in exact terms?