Adding Sinusoids of differing Magnitute & Phases

In summary, to solve for A and \theta in the equation Acos(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t), you need to use the Pythagorean theorem to find the magnitude of A and inverse tangent to find the angle \theta. This is because the left side of the equation represents a vector with components of cosine and sine, while the right side represents the x and y components of the same vector. The values of A and \theta will not change if you write the equation as Asin(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t).
  • #1
jeff1evesque
312
0

Homework Statement


Find A and [tex]\theta[/tex] given that:
[tex]Acos(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t)[/tex]
Could someone elaborate on how to solve this. I mean it looks to me that one simply takes the magnitude of the coefficient and the inverse tangent of the same coefficients. But I feel there needs to be justification as to why we're allowed to do this.

Homework Equations


not sure.

The Attempt at a Solution


But the solution (according to my notes) is: [tex]A = \sqrt{4^2 + 3^2} = 5[/tex] and [tex]\theta^{-1}[/tex] = [tex]\frac{4}{3} = 53.1[/tex]

Thanks,

JL
 
Last edited:
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  • #2
What if instead of

[tex]
Acos(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t)
[/tex],

I wanted to write it as,

[tex]
Asin(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t)
[/tex],

would the values of A and [tex]\theta[/tex] change at all?
 
  • #3
jeff1evesque said:
What if instead of

[tex]
Acos(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t)
[/tex],

I wanted to write it as,

[tex]
Asin(\omega t + \theta) = 4sin(\omega t) + 3 cos(\omega t)
[/tex],

would the values of A and [tex]\theta[/tex] change at all?
Oh, actually I think I know. The x coordinate axis is [tex]Acos(\omega t)[/tex] (negative for the negative x-axis), and the y coordinate axis is [tex]Asin(\omega t)[/tex] (negative for the negative y-axis). So this means the left side value in the equality is the actual vector, and the terms on the right are the x and y components.

Thannks,JL
 
  • #4
You need addition and subtraction formulas for sine and cosine (or you can use Euler's identity, but that is more advanced).
 

1. What is the purpose of adding sinusoids of differing magnitude and phases?

The purpose of adding sinusoids of differing magnitude and phases is to study the phenomenon of interference, where waves combine to form a resultant wave. This can help us understand how sound and light waves behave in different situations.

2. How do you calculate the resultant amplitude when adding sinusoids of differing magnitude and phases?

To calculate the resultant amplitude, you can use the Pythagorean theorem, where the resultant amplitude is equal to the square root of the sum of the squares of the individual amplitudes. This can also be represented using vector addition.

3. Can adding sinusoids of differing magnitude and phases cancel each other out?

Yes, it is possible for adding sinusoids of differing magnitude and phases to cancel each other out. This occurs when the two waves have equal but opposite amplitudes and are perfectly out of phase. This is known as destructive interference, where the resultant wave has an amplitude of zero.

4. How do the phases of the sinusoids affect the interference pattern?

The phases of the sinusoids determine whether the interference is constructive or destructive. When the two waves are in phase, they add up to create a larger amplitude, known as constructive interference. When they are out of phase, they cancel each other out, resulting in a smaller or zero amplitude, known as destructive interference.

5. What are some real-life examples of adding sinusoids of differing magnitude and phases?

Real-life examples of adding sinusoids of differing magnitude and phases include sound waves from different musical instruments, where the combination of waves creates a unique sound. Another example is light waves from different sources, such as lasers, which can create interference patterns known as diffraction patterns.

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