Adding sinusoidal functions using phasors

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Discussion Overview

The discussion revolves around the addition of sinusoidal functions using phasors, focusing on the calculation of angles in the complex plane and the use of trigonometric functions to determine these angles. Participants explore the implications of using different methods for angle calculation and the potential discrepancies that arise from them.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a calculation involving phasors and expresses confusion over the angle obtained compared to the textbook's answer, questioning how the correct angle is determined.
  • Another participant suggests that phasors may need to be converted to RMS before performing calculations, indicating a potential oversight in the initial approach.
  • A participant explains that the arctan function does not account for the quadrant in which the angle lies, suggesting the use of a sketch on the complex plane to determine the correct angle.
  • It is noted that calculators may have functions like atan2() that can provide the correct angle by considering both the x and y components, which could help avoid quadrant-related errors.
  • One participant acknowledges their forgetfulness regarding trigonometric principles and expresses relief at understanding the concept again.
  • Several participants assert that the calculated value of -1.376 + 3.02j does not correspond to -65 degrees, insisting that it should equal 114.49 degrees, prompting further examination of the calculations involved.
  • There is a discussion about the limitations of the arctan function, which returns principal values that may not reflect the actual angle in the context of the problem, leading to confusion in angle determination.
  • Participants suggest using both trigonometric methods and calculator functions to verify results, emphasizing the importance of consistency in calculations.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correct angle derived from the calculations, with some insisting on the accuracy of 114.49 degrees while others report obtaining -65 degrees. The discussion remains unresolved as different methods yield conflicting results.

Contextual Notes

Participants highlight the importance of quadrant considerations when using inverse trigonometric functions, as well as the potential for calculators to return principal values that may not align with the expected results in this context.

Vishera
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Homework Statement



tgiXxOl.png


Homework Equations


The Attempt at a Solution


$$3cos(20t+10°)-5cos(20t-30°)\\ =3\angle 10°-5\angle -30°\\ =-1.376+3.0209j\\ =3.32\angle -65.51°$$ In the last step, the textbook actually got ##3.32\angle 114.49°##.

I checked both answers and it seems that the textbook's answer is correct. All I did to get -65.51° is using the arctan function on my calculator. It looks like their angle is 180° greater than that. How did they choose the correct angle? I'm rusty on my trigonometry.
 
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Don't phasors need to be put in RMS before you work with them in this case?
 
Last edited:
Vishera said:

Homework Statement



tgiXxOl.png


Homework Equations





The Attempt at a Solution


$$3cos(20t+10°)-5cos(20t-30°)\\ =3\angle 10°-5\angle -30°\\ =-1.376+3.0209j\\ =3.32\angle -65.51°$$ In the last step, the textbook actually got ##3.32\angle 114.49°##.

I checked both answers and it seems that the textbook's answer is correct. All I did to get -65.51° is using the arctan function on my calculator. It looks like their angle is 180° greater than that. How did they choose the correct angle? I'm rusty on my trigonometry.

The arctan() function doesn't discriminate between the cases where the minus sign is to be associated with the numerator or the denominator of the argument; it only sees the overall sign of the single number it is given. To check what quadrant the correct angle should lie in, do a quick sketch of the x (real) and y (imaginary) values on the complex plane. Adjust the result by 180° if required.

If your calculator has an atan2() function, use that instead as it takes two arguments, one for the x component and one for the y, and sorts out the correct angle automatically. Failing that, your calculator might have rectangular to polar conversion built-in, which will also give the result in the correct quadrant automatically.
 
Draw a picture of your combined phasor. See the quadrant where the components locate the resultant?

Generally, the inverse trig functions on your calculator only return the principal angles, which for the tangent is in the range -π/2 ≤ θ ≤ π/2. When doing these types of calculations, you must give your result a separate check to make sure you obtain the correct angle.
 
Thanks for help guys. I completely forgot this part of trigonometry and I get it now. It's actually kind of scary that I forgot this because it's not like I stopped doing Math. Too much calculus has made me a bit rusty on my algebra (or at least trigonometry).
 
-1.376 + 3.02J never equals -65 degrees...check your math there...it always equals 114.49 degrees.
 
psparky said:
-1.376 + 3.02J never equals -65 degrees...check your math there...it always equals 114.49 degrees.

Weird. My calculator is always giving me a principal value of -65.
 
Vishera said:
Weird. My calculator is always giving me a principal value of -65.

Yes. Because you are calculating:
$$\phi = arctan\left( \frac{3.02}{-1.376} \right)$$
Which, after resolving the argument to -2.195, the arctan() function places in the fourth quadrant. It cannot distinguish ##\frac{3.02}{-1.376}## from ##\frac{-3.02}{1.376}## or ##-2.195##, and can only return a corresponding angle that lies within its principal range of +/- 90° .
 
gneill said:
Yes. Because you are calculating:
$$\phi = arctan\left( \frac{3.02}{-1.376} \right)$$
Which, after resolving the argument to -2.195, the arctan() function places in the fourth quadrant. It cannot distinguish ##\frac{3.02}{-1.376}## from ##\frac{-3.02}{1.376}## or ##-2.195##, and can only return a corresponding angle that lies within its principal range of +/- 90° .

Trig is certainly one way, another is simply using the rectangular to polar function on your calculator.
Or do them both. If they both agree, better yet.
 

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