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Find what is wrong with the students current answer 

The student's solution involves converting the rectangular coordinates \((R_x, R_y)\) to polar coordinates \((R, \Theta)\). Let's walk through their work and check for any mistakes:

### 1. Magnitude \( R \):
The magnitude \( R \) is given by the formula:
\[
R = \sqrt{R_x^2 + R_y^2}
\]

Using the values:
- \( R_x = -0.271 \)
- \( R_y = 6.624 \)

The student correctly calculated:
\[
R = \sqrt{(-0.271)^2 + (6.624)^2} = \sqrt{43.8039} \approx 6.6185
\]
This part of the solution is correct.

### 2. Angle \( \Theta \):
The angle \( \Theta \) is found using the inverse tangent:
\[
\tan(\Theta) = \frac{R_y}{R_x} = \frac{6.624}{-0.271}
\]
This results in:
\[
\tan(\Theta) = -24.45
\]

The student calculated:
\[
\Theta = \arctan(-24.45) \approx -87.66^\circ
\]

However, because the point \((-0.271, 6.624)\) lies in the **second quadrant** (since \( R_x \) is negative and \( R_y \) is positive), the angle should be adjusted by adding \(180^\circ\) to the result of the arctangent. The correct calculation is:
\[
\Theta = -87.66^\circ + 180^\circ = 92.34^\circ
\]

### Conclusion:
The student's mistake lies in their final step of adjusting the angle. They incorrectly added \(360^\circ\) instead of the correct quadrant adjustment of \(180^\circ\). The correct angle is:
\[
\Theta = 92.34^\circ
\]
instead of \(272.34^\circ\).

Would you like to explore this solution further?

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### Follow-up Questions:
1. How would the result change if \( R_x \) and \( R_y \) were both positive?
2. Can you calculate the angle \( \Theta \) for a point in the third quadrant?
3. How does changing the sign of \( R_y \) impact the calculation of \( \Theta \)?
4. What are the alternative ways to find \( \Theta \) without using the arctangent function?
5. What is the significance of the angle correction when converting between Cartesian and polar coordinates?

**Tip:** Always check the quadrant of your point before finalizing the angle when converting from Cartesian to polar coordinates.

Find R and Θ given the components Rx = -0.271 and Ry = 6.624.

Learn how to convert rectangular coordinates Rx = -0.271 and Ry = 6.624 to polar form by calculating the magnitude and angle. This detailed solution covers key trigonometric concepts, including Pythagoras' theorem and quadrant adjustments for angles. Suitable for students in Grades 10-12.

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