d) The voltages $$ V_{1} $$ and $$ V_{2} $$ are represented by waveforms as follows $$ V_{1}=9 \sin (\omega t) $$ $$ V_{2}=2 \cos (\omega t) $$ If $$ V_{3} $$ is the sum of $$ V_{1} $$ and $$ V_{2} $$ such that $$ V_{3}=V_{1}+V_{2} $$ Find the expression of $$ V_{3} $$ in sine form $$ V_{3}=R \sin (\omega t+\alpha) $$ where $$ \alpha $$ is expressed in radians. Both figures to two decimal places. $$ V_{3}=\square \sin (\omega t+\square) $$ rks red s. submit part

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To find the expression for $$ V_{3} $$ in the form $$ V_{3} = R \sin(\omega t + \alpha) $$, we will follow these steps:

1. **Express $$ V_{1} $$ and $$ V_{2} $$ in a common form**: 
   - We have $$ V_{1} = 9 \sin(\omega t) $$ and $$ V_{2} = 2 \cos(\omega t) $$.
   - We can convert $$ V_{2} $$ into sine form using the identity $$ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) $$.

2. **Combine the two waveforms**:
   - Rewrite $$ V_{2} $$:
     $$
     V_{2} = 2 \cos(\omega t) = 2 \sin\left(\omega t + \frac{\pi}{2}\right)
     $$
   - Now we can express $$ V_{3} $$:
     $$
     V_{3} = V_{1} + V_{2} = 9 \sin(\omega t) + 2 \sin\left(\omega t + \frac{\pi}{2}\right)
     $$

3. **Use the sine addition formula**:
   - We can express $$ V_{3} $$ in the form $$ R \sin(\omega t + \alpha) $$ using the formula:
     $$
     R = \sqrt{A^2 + B^2}
     $$
     where $$ A $$ and $$ B $$ are the coefficients of $$ \sin(\omega t) $$ and $$ \sin\left(\omega t + \frac{\pi}{2}\right) $$ respectively.

4. **Calculate $$ R $$ and $$ \alpha $$**:
   - Here, $$ A = 9 $$ and $$ B = 2 $$.
   - Calculate $$ R $$:
     $$
     R = \sqrt{9^2 + 2^2} = \sqrt{81 + 4} = \sqrt{85}
     $$
   - Calculate $$ \alpha $$ using:
     $$
     \tan(\alpha) = \frac{B}{A} = \frac{2}{9}
     $$
   - Therefore, $$ \alpha = \tan^{-1}\left(\frac{2}{9}\right) $$.

Now, let's calculate $$ R $$ and $$ \alpha $$ using the appropriate tool calls.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{85}$$
Calculate or simplify the expression $$ \arctan(2/9) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\arctan\left(\frac{2}{9}\right)$$
Now we have the results for $$ R $$ and $$ \alpha $$:

1. **Magnitude $$ R $$**:
   $$
   R \approx 9.22
   $$

2. **Angle $$ \alpha $$**:
   $$
   \alpha \approx 0.22 \text{ radians}
   $$

Thus, we can express $$ V_{3} $$ in the desired form:

$$
V_{3} \approx 9.22 \sin(\omega t + 0.22)
$$

Finally, we can fill in the blanks:

$$
V_{3} = 9.22 \sin(\omega t + 0.22)
$$
$$ V_{3} = 9.22 \sin(\omega t + 0.22) $$

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