https://file.aiquickdraw.com/mathbot/file/17290928538503u0ycxj2.jpg

The question in the image asks to convert the vector \((3, 4)\) from Cartesian (rectangular) coordinates to polar form. Here's how to solve it step-by-step:

1. **Magnitude (r)**: The magnitude of the vector is calculated using the Pythagorean theorem:
   \[
   r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
   \]

2. **Angle (θ)**: The angle θ is found using the inverse tangent function:
   \[
   \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ
   \]

So, the polar form of the vector \((3, 4)\) is:
\[
(5, 53.13^\circ)
\]

### Answer: The correct option is (5, 53.13°).

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For **Question 9**, it asks for the period of a sinusoidal wave with a frequency of 2 Hz. The period \( T \) of a wave is the reciprocal of its frequency \( f \):
\[
T = \frac{1}{f}
\]
Given \( f = 2 \, \text{Hz} \), the period is:
\[
T = \frac{1}{2} = 0.5 \, \text{seconds}
\]

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Let me know if you'd like any additional explanations or details!

Here are 5 related questions for further practice:
1. How do you convert a polar vector back to Cartesian coordinates?
2. What is the polar form of the vector \((-3, 4)\)?
3. How would the vector \((3, 4)\) look if rotated 45 degrees counterclockwise?
4. What is the frequency of a wave with a period of 0.25 seconds?
5. How does changing the frequency of a wave affect its wavelength?

### Tip: When converting vectors, always remember to check if the angle is in the correct quadrant based on the signs of \(x\) and \(y\).

Learn how to convert the vector (3, 4) from Cartesian coordinates to polar form. This problem covers vector magnitude and trigonometric angle calculations. Perfect for students in Grades 10-12.

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