https://file.aiquickdraw.com/mathbot/file/1726126563584un2dn2ql.jpg

Let's work through the problems in the image.

### Problem b)
We are given:
- \(\phi = 41.3^\circ\)
- \(v_L = 6.24 \, V\)

We need to calculate \(v_R\).

Using the following relation from AC circuit theory:

\[
\tan(\phi) = \frac{v_L}{v_R}
\]

We can rearrange this to solve for \(v_R\):

\[
v_R = \frac{v_L}{\tan(\phi)}
\]

### Problem c)
We are given:
- \(\phi = 27.5^\circ\)
- \(v_R = 23.8 \, V\)

We need to calculate \(v\), the total voltage.

Using the relationship between \(v_R\), \(v_L\), and \(v\) in an AC circuit:

\[
v = \sqrt{v_R^2 + v_L^2}
\]

We can first calculate \(v_L\) using:

\[
\tan(\phi) = \frac{v_L}{v_R}
\]

Then substitute \(v_L\) back into the equation for \(v\).

### Problem d)
We are given:
- \(v = 126 \, V\)
- \(v_R = 89.3 \, V\)

We need to calculate \(\phi\) in degrees and minutes.

Using the following relationship:

\[
\cos(\phi) = \frac{v_R}{v}
\]

Rearranging to solve for \(\phi\):

\[
\phi = \cos^{-1}\left(\frac{v_R}{v}\right)
\]

Finally, we can convert the decimal part of the degrees into minutes.

Let me calculate these values step by step.Here are the solutions for the given problems:

- **Problem b**: The value of \( v_R \) is approximately \( 7.10 \, V \).
- **Problem c**: The total voltage \( v \) is approximately \( 26.83 \, V \).
- **Problem d**: The phase angle \( \phi \) is approximately \( 44^\circ \, 52' \) (44 degrees and 52 minutes).

Would you like further details or clarifications on these calculations?

Here are five related questions for further exploration:
1. How does the phase angle \(\phi\) affect the relationship between \(v_L\) and \(v_R\)?
2. What would happen if the phase angle \(\phi\) were 90 degrees?
3. How can the total voltage \(v\) be derived from complex impedance in an AC circuit?
4. Can the angle \(\phi\) be negative, and what does it signify in AC circuit theory?
5. How would you calculate \(v_L\) directly if given the total voltage \(v\) and phase angle \(\phi\)?

**Tip**: Always use radians in trigonometric functions when working with angles in calculations!

Given that ϕ = 41.3° and vL = 6.24 V, evaluate vR. Given that ϕ = 27.5° and vR = 23.8 V, evaluate v. Given that v = 126 V and vR = 89.3 V, evaluate ϕ in degrees and minutes.

This problem explores how to evaluate the voltage components vR, total voltage v, and phase angle ϕ in an AC circuit using trigonometric relationships. It covers key formulas such as tan(ϕ) = vL / vR, v = √(vR^2 + vL^2), and cos(ϕ) = vR / v.

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