VIPSolutionsAct like a helpful tutor and exlain me :3. Given the £(1) defined below, express z(t) in the form 2(t) = Acos(uwt + 6), (t=) Aueos(2e ut + 6x) where Ay = 5, Az =9, Ay =2,61= }r, 62 = Lx, 6 = 0, ane fi = f2 = fo. Show all your steps.
Question:
Act like a helpful tutor and exlain me :
3. Given the £(1) defined below, express z(t) in the form 2(t) = Acos(uwt + 6),
(t=) Aueos(2e ut + 6x)
where Ay = 5, Az =9, Ay =2,61= }r, 62 = Lx, 6 = 0, ane fi = f2 = fo. Show all your steps.
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Asked by: Alok Kumar
Created at: 2026-01-27 08:53:41
Asked by: Alok Kumar
Created at: 2026-01-27 08:53:41
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Let’s walk through this problem step-by-step!
---
## **Given**
You are given:
\[
x(t) = \sum_{k=1}^{3} A_k \cos(2\pi f_k t + \phi_k)
\]
with parameters:
- \( A_1 = 5 \), \( A_2 = 9 \), \( A_3 = 2 \)
- \( \phi_1 = \frac{1}{4}\pi \), \( \phi_2 = \frac{1}{8}\pi \), \( \phi_3 = \)
- \( f_1 = f_2 = f_3 = f \)
You are asked to write \( x(t) \) as:
\[
x(t) = A \cos(\omega t + \phi)
\]
where \( \omega = 2\pi f \).
---
## **Step 1: Expand the Sum**
Since all frequencies are equal, set \( \omega = 2\pi f \):
\[
x(t) = 5\cos(\omega t + \frac{\pi}{4}) + 9\cos(\omega t + \frac{\pi}{8}) + 2\cos(\omega t)
\]
---
## **Step 2: Express Each Cosine in Terms of \(\cos(\omega t)\) and \(\sin(\omega t)\)**
Recall:
\[
\cos(\omega t + \phi) = \cos(\phi)\cos(\omega t) - \sin(\phi)\sin(\omega t)
\]
Apply to each term:
- \( 5\cos(\omega t + \frac{\pi}{4}) = 5\left[\cos(\frac{\pi}{4})\cos(\omega t) - \sin(\frac{\pi}{4})\sin(\omega t)\right] \)
- \( 9\cos(\omega t + \frac{\pi}{8}) = 9\left[\cos(\frac{\pi}{8})\cos(\omega t) - \sin(\frac{\pi}{8})\sin(\omega t)\right] \)
- \( 2\cos(\omega t) = 2\cos(\omega t) \) (since cosine of is 1, sine is )
---
## **Step 3: Combine Terms**
Group all \(\cos(\omega t)\) and \(\sin(\omega t)\) coefficients:
\[
x(t) = [5\cos(\frac{\pi}{4}) + 9\cos(\frac{\pi}{8}) + 2] \cos(\omega t) - [5\sin(\frac{\pi}{4}) + 9\sin(\frac{\pi}{8})]\sin(\omega t)
\]
Let:
- \( C = 5\cos(\frac{\pi}{4}) + 9\cos(\frac{\pi}{8}) + 2 \)
- \( S = 5\sin(\frac{\pi}{4}) + 9\sin(\frac{\pi}{8}) \)
So,
\[
x(t) = C\cos(\omega t) - S\sin(\omega t)
\]
---
## **Step 4: Write as a Single Cosine**
Recall:
\[
C\cos(\omega t) - S\sin(\omega t) = A\cos(\omega t + \phi)
\]
where:
- \( A = \sqrt{C^2 + S^2} \)
- \( \phi = -\arctan\left(\frac{S}{C}\right) \)
---
## **Step 5: Plug in Values**
- \( \cos(\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \approx .7071 \)
- \( \cos(\frac{\pi}{8}) \approx .9239 \)
- \( \sin(\frac{\pi}{8}) \approx .3827 \)
Calculate:
\[
C = 5 \times .7071 + 9 \times .9239 + 2 = 3.5355 + 8.3151 + 2 = 13.8506
\]
\[
S = 5 \times .7071 + 9 \times .3827 = 3.5355 + 3.4443 = 6.9798
\]
---
## **Step 6: Find \(A\) and \(\phi\)**
\[
A = \sqrt{C^2 + S^2} = \sqrt{(13.8506)^2 + (6.9798)^2} \approx \sqrt{192.02 + 48.73} \approx \sqrt{240.75} \approx 15.51
\]
\[
\phi = -\arctan\left(\frac{S}{C}\right) = -\arctan\left(\frac{6.9798}{13.8506}\right) \approx -.464 \text{ radians}
\]
---
## **Final Answer**
\[
\boxed{
x(t) = 15.51 \cos(\omega t - .464)
}
\]
where \( \omega = 2\pi f \).
---
### **Summary of Steps**
1. **Expand** the sum using cosine addition.
2. **Group** \(\cos(\omega t)\) and \(\sin(\omega t)\) terms.
3. **Calculate** the coefficients \(C\) and \(S\).
4. **Convert** to single cosine form using amplitude \(A\) and phase \(\phi\).
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