The questions in the uploaded exercise involve polar coordinates, transformations, geometry, and trigonometry. Below are some key solutions and explanations for specific problems in the exercise:

1. Find the polar coordinates of the given Cartesian points:

• The conversion formulas are: $r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)$

Example (i): Cartesian point $(\sqrt{3}, 1)$

• $r = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2$
• $\theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}$

Polar coordinates: $(2, \frac{\pi}{6})$.

I can solve the others if needed.

2. Find the polar distance between the points:

• Use the formula: $d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_2 - \theta_1)}$

Example: $(3, 45^\circ)$ and $(7, 105^\circ)$

• Convert $\theta$ to radians: $45^\circ = \frac{\pi}{4}, 105^\circ = \frac{7\pi}{12}$
• Substitute $r_1 = 3, r_2 = 7, \theta_1 = \frac{\pi}{4}, \theta_2 = \frac{7\pi}{12}$.

If required, I can solve this step-by-step for clarity.

3. Determine the nature of the triangle and its area:

• Vertices: (0, 0), (3, π/6), and (4, π/3).
• Convert all polar coordinates to Cartesian coordinates, then use the distance formula to find sides: $x = r\cos\theta, \quad y = r\sin\theta$
• For area, use: $\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|$

4. Find the area of a triangle using angular polar points:

• For points $(r_1, \theta_1), (r_2, \theta_2), (r_3, \theta_3)$: $\text{Area} = \frac{1}{2} \left| r_1r_2\sin(\theta_2-\theta_1) + r_2r_3\sin(\theta_3-\theta_2) + r_3r_1\sin(\theta_1-\theta_3) \right|$
• Example: Points $(6, \frac{\pi}{3}), (3, \frac{8\pi}{3}), (3, \frac{\pi}{3})$.

5. Transform the equations to polar form:

• Conversion rules: $x = r\cos\theta, \quad y = r\sin\theta, \quad x^2 + y^2 = r^2$
• Example: For $x^2 + y^2 = 2xy$, substitute: $r^2 = 2r^2\cos\theta\sin\theta$
• Simplify: $r = 2\sin(2\theta)$

Let me know which specific part you want fully solved with detailed steps.

5 Questions for Further Exploration:

1. Can you explain the geometrical interpretation of polar distance?
2. How do angular measures impact the triangle's area in polar form?
3. What are the applications of transforming equations into polar coordinates?
4. Why is the polar coordinate system preferable in certain problems over Cartesian?
5. How can the locus equation for the last problem be derived geometrically?

Tip:

Always double-check angular conversions (degrees to radians) when working in polar coordinates to ensure accuracy.