QuEsTION $$ 2(14 $$ marks) The current $$ / $$ (in $$ A) $$ at time $$ t $$ (in seconds) in a particular circuit is given by $$ i=3 \sin t+4 \cos t . $$ a. If $$ / $$ can be expressed in the form $$ R \sin (t+\phi) $$, find $$ R $$ and $$ \phi $$. b. State the maximum value of $$ i $$. G. Find the first positive time $$ t $$, when $$ i=4.5 \mathrm{~A} $$.

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Let's solve each part of the problem step-by-step.

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**Given:**
$$ i(t) = 3 \sin t + 4 \cos t $$

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### **a. Expressing $$ i(t) $$ in the Form $$ R \sin(t + \phi) $$**

We aim to rewrite $$ i(t) $$ as:
$$ i(t) = R \sin(t + \phi) $$

Using the sine addition formula:
$$ R \sin(t + \phi) = R \sin t \cos \phi + R \cos t \sin \phi $$

Comparing this with the original expression:
$$ 3 \sin t + 4 \cos t = R \sin t \cos \phi + R \cos t \sin \phi $$

**Equating Coefficients:**
$$
\begin{cases}
R \cos \phi = 3 \quad &(1) \
R \sin \phi = 4 \quad &(2)
\end{cases}
$$

**Solving for $$ R $$ and $$ \phi $$:**

1. **Find $$ R $$:**
   $$
   R = \sqrt{(R \cos \phi)^2 + (R \sin \phi)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
   $$

2. **Find $$ \phi $$:**
   $$
   \tan \phi = \frac{R \sin \phi}{R \cos \phi} = \frac{4}{3} \
   \Rightarrow \phi = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ \quad \text{or} \quad 0.927 \text{ radians}
   $$

**Final Expression:**
$$
i(t) = 5 \sin(t + 0.927) \quad \text{or} \quad i(t) = 5 \sin\left(t + \tan^{-1}\left(\frac{4}{3}\right)\right)
$$

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### **b. Maximum Value of $$ i(t) $$**

The general form $$ R \sin(t + \phi) $$ has a maximum value equal to $$ R $$.

Given $$ R = 5 $$:

$$
\text{Maximum value of } i(t) = 5 \, \text{A}
$$

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### **c. Finding the First Positive Time $$ t $$ When $$ i = 4.5 \, \text{A} $$**

**Given:**
$$
i(t) = 5 \sin(t + \phi) = 4.5
$$

1. **Solve for $$ \sin(t + \phi) $$:**
   $$
   \sin(t + \phi) = \frac{4.5}{5} = 0.9
   $$

2. **Find $$ t + \phi $$:**
   $$
   t + \phi = \sin^{-1}(0.9) \approx 1.119 \, \text{radians}
   $$
   
3. **Subtract $$ \phi $$ to find $$ t $$:**
   $$
   t = 1.119 - 0.927 = 0.192 \, \text{seconds}
   $$

**Therefore, the first positive time when $$ i = 4.5 \, \text{A} $$ is approximately $$ t = 0.192 \, \text{seconds} $$.**

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**Summary of Answers:**

a. $$ R = 5 $$ and $$ \phi = \tan^{-1}\left(\frac{4}{3}\right) $$ radians (approximately $$ 0.927 $$ radians).

b. The maximum value of $$ i $$ is $$ 5 $$ A.

c. The first positive time $$ t $$ when $$ i = 4.5 $$ A is approximately $$ 0.192 $$ seconds.
a. $$ R = 5 $$ and $$ \phi \approx 0.927 $$ radians. b. The maximum value of $$ i $$ is $$ 5 $$ A. c. The first positive time $$ t $$ when $$ i = 4.5 $$ A is approximately $$ 0.192 $$ seconds.

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