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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, plus, b, i\mid, z, \mid, equals, square root of, a, squared, plus, b, squared, end square root
Angle of a, plus, b, itheta, equals, tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, b, divided by, a, end fraction, right parenthesis
Rectangular form from absolute value r and angle thetar, cosine, left parenthesis, theta, right parenthesis, plus, r, sine, left parenthesis, theta, right parenthesis, dot, 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 start color #11accd, start text, r, e, a, l, end text, end color #11accd and start color #1fab54, start text, i, m, a, g, i, n, a, r, y, end text, end color #1fab54 parts. For example, start color #11accd, 3, end color #11accd, plus, start color #1fab54, 4, end color #1fab54, i.
We can plot numbers in the complex plane according to their parts:
Considered graphically, there's another way to uniquely describe complex numbers — their start color #e07d10, start text, a, b, s, o, l, u, t, e, space, v, a, l, u, e, end text, end color #e07d10 and start color #aa87ff, start text, a, n, g, l, e, end text, end color #aa87ff:
The start color #e07d10, start text, a, b, s, o, l, u, t, e, space, v, a, l, u, e, end text, end color #e07d10, or start color #e07d10, start text, m, o, d, u, l, u, s, end text, end color #e07d10, gives the distance of the number from the origin in the complex plane, while its start color #aa87ff, start text, a, n, g, l, e, end text, end color #aa87ff, or start color #aa87ff, start text, a, r, g, u, m, e, n, t, end text, end color #aa87ff, 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, vertical bar, z, vertical bar.
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):
vertical bar, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, i, vertical bar, equals, square root of, start color #11accd, a, end color #11accd, squared, plus, start color #1fab54, b, end color #1fab54, squared, end square root
For example, the absolute value of start color #11accd, 3, end color #11accd, plus, start color #1fab54, 4, end color #1fab54, i is square root of, start color #11accd, 3, end color #11accd, squared, plus, start color #1fab54, 4, end color #1fab54, squared, end square root, equals, square root of, 25, end square root, equals, 5.
Problem 1.1
vertical bar, 3, plus, 7, i, vertical bar, equals

Give an exact answer.

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, equals, tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, start color #1fab54, b, end color #1fab54, divided by, start color #11accd, a, end color #11accd, end fraction, right parenthesis
This results from using trigonometry in the right triangle formed by the number and the Real axis.

Example 1: Quadrant start text, I, end text

Let's find the angle of start color #11accd, 3, end color #11accd, plus, start color #1fab54, 4, end color #1fab54, i:
tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, start color #1fab54, 4, end color #1fab54, divided by, start color #11accd, 3, end color #11accd, end fraction, right parenthesis, approximately equals, 53, degrees

Example 2: Quadrant start text, I, I, end text

Let's find the angle of start color #11accd, minus, 3, end color #11accd, plus, start color #1fab54, 4, end color #1fab54, i. First, notice that start color #11accd, minus, 3, end color #11accd, plus, start color #1fab54, 4, end color #1fab54, i is in Quadrant start text, I, I, end text.
tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, start color #1fab54, 4, end color #1fab54, divided by, start color #11accd, minus, 3, end color #11accd, end fraction, right parenthesis, approximately equals, minus, 53, degrees
minus, 53, degrees is in Quadrant start text, I, V, end text, not start text, I, I, end text. We must add 180, degrees to obtain the opposite angle:
minus, 53, degrees, plus, 180, degrees, equals, 127, degrees
Problem 2.1
z, equals, 1, plus, 4, i
theta, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees
Round your answer, if necessary, to the nearest tenth. Express theta between minus, 180, degrees and 180, degrees.

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:
start overbrace, start color #e07d10, r, end color #e07d10, cosine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, end overbrace, start superscript, start color #11accd, a, end color #11accd, end superscript, plus, start overbrace, start color #e07d10, r, end color #e07d10, sine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, end overbrace, start superscript, start color #1fab54, b, end color #1fab54, end superscript, dot, 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 start color #e07d10, 2, end color #e07d10 and angle is start color #aa87ff, 30, degrees, end color #aa87ff:
start color #e07d10, 2, end color #e07d10, cosine, left parenthesis, start color #aa87ff, 30, degrees, end color #aa87ff, right parenthesis, plus, start color #e07d10, 2, end color #e07d10, sine, left parenthesis, start color #aa87ff, 30, degrees, end color #aa87ff, right parenthesis, i, equals, start color #11accd, square root of, 3, end square root, end color #11accd, plus, start color #1fab54, 1, end color #1fab54, i
Problem 3.1
vertical bar, z, start subscript, 1, end subscript, vertical bar, equals, 3 and theta, start subscript, 1, end subscript, equals, 20, degrees
z, start subscript, 1, end subscript, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
+
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
i
Round your answers to the nearest thousandth.

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

Want to join the conversation?

  • hopper cool style avatar for user 🤔 ᴄᴏᴅᴇᴅ ɢᴇɴɪᴜȿ 😎
    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?
    (14 votes)
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    • purple pi pink style avatar for user ZeroFK
      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|.
      (9 votes)
  • blobby green style avatar for user marialagakos
    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)?
    (12 votes)
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    • blobby green style avatar for user matthewmorris628
      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.

      Hope this is helpful!
      (9 votes)
  • starky ultimate style avatar for user nmj834481993
    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).
    (6 votes)
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    • aqualine ultimate style avatar for user Sarah
      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 :)
      (6 votes)
  • blobby green style avatar for user dsumner602.DS
    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?
    (6 votes)
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  • aqualine ultimate style avatar for user anahope03
    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?
    (6 votes)
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  • male robot donald style avatar for user Akshat
    Can someone explain how to find inverse of trigo functions
    (2 votes)
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  • leaf green style avatar for user at155204
    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.
    (3 votes)
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    • mr pink red style avatar for user andrewp18
      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!
      (3 votes)
  • hopper cool style avatar for user Wilson Cheung
    Why do we need to add 180 degrees in Example 2: Quadrant ||
    (3 votes)
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    • female robot grace style avatar for user El
      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?
      (3 votes)
  • blobby green style avatar for user Pradeep Kumar
    how do we know when to subtract 180 degrees from theta and when i should add 180 degrees to theta? Please help!
    (3 votes)
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  • cacteye yellow style avatar for user Br Paul
    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?
    (2 votes)
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    • female robot grace style avatar for user loumast17
      So you need your unit circle memorized or written down.

      You will always be given the absolute value of the complex number (the length) and angle the line makes with the positive x axis. Let's call the length r and angle t.

      The first step is easy, you take those two numbers and plug them into this.

      r*cos(t) + r*sin(t)*i

      From there you solve the trig functions, which is where you ned your unit circle.

      Does that help?
      (2 votes)