Question 1

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) × |*z*₁|`
`b = sin(θ₁) × |*z*₁|`

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?

Accepted Answer

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|.

Question 2

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)?

Accepted Answer

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!

Question 3

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).

Accepted Answer

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 :)

Question 4

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.

Accepted Answer

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!

Question 5

Why do we need to add 180 degrees in Example 2: Quadrant ||

Accepted Answer

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?

Question 6

how do we know when to subtract 180 degrees from theta and when i should add 180 degrees to theta? Please help!

Accepted Answer

If the angle theta lies in quadrant 2 you must add 180 and quadrant 3 substract 180

Question 7

Can someone explain how to find inverse of trigo functions

Accepted Answer

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)).

Question 8

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?

Accepted Answer

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?


Absolute value of $a + b i$ ‍		$∣ z ∣= \sqrt{a^{2} + b^{2}}$ ‍
Angle of $a + b i$ ‍		$θ = \tan^{- 1} (\frac{b}{a})$ ‍
Rectangular form from absolute value $r$ ‍ and angle $θ$ ‍		$r \cos (θ) + r \sin (θ) \cdot i$ ‍

Precalculus

Course: Precalculus > Unit 3

Complex number absolute value & angle review

What are the absolute value and angle of complex numbers?

Practice set 1: Finding absolute value

Practice set 2: Finding angle

Example 1: Quadrant $I$ ‍

Example 2: Quadrant $II$ ‍

Practice set 3: Rectangular form from absolute value and angle

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