Here’s a detailed explanation and solution to each of the problems in the image:

6. Simplify: cot(sin⁻¹(2/3))

1. Let $\theta = \sin^{-1}(2/3)$. Then $\sin(\theta) = 2/3$.
2. Use the Pythagorean identity: $\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - (2/3)^2} = \sqrt{1 - 4/9} = \sqrt{5/9} = \sqrt{5}/3$.
3. Cotangent is given by: $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\sqrt{5}/3}{2/3} = \frac{\sqrt{5}}{2}$.

Answer: (a) $\frac{\sqrt{5}}{2}$

7. Solve the system of equations: 2x + 5y = 8, 6x + 15y = 24

1. Rewrite the second equation: $6x + 15y = 24 \rightarrow 3(2x + 5y) = 24$, which simplifies to $2x + 5y = 8$, identical to the first equation.
2. Since the two equations are identical, the system has infinitely many solutions.

Answer: (b) Infinitely many solutions

8. Convert rectangular coordinates (1, -1) into polar coordinates

1. Polar coordinates $(r, \theta)$ are given by:
   • $r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-1)^2} = \sqrt{2}$,
   • $\theta = \tan^{-1}(y/x) = \tan^{-1}(-1/1) = \tan^{-1}(-1)$.
2. Since the point is in Quadrant IV, $\theta = \frac{7\pi}{4}$ (adjusted for quadrant).

Answer: (b) $(- \sqrt{2}, \frac{7\pi}{4})$

9. Find the rectangular coordinates for the point whose polar coordinates are (5, $5\pi$)

1. Rectangular coordinates are given by:
   • $x = r \cos(\theta) = 5 \cos(5\pi)$,
   • $y = r \sin(\theta) = 5 \sin(5\pi)$.
2. Since $\cos(5\pi) = -1$ and $\sin(5\pi) = 0$,
   • $x = 5(-1) = -5$,
   • $y = 0$.

Answer: (e) $(-5, 0)$

10. Which point is NOT the same as the polar coordinates $(-3, \frac{5\pi}{3})$?

1. $(-3, \frac{5\pi}{3})$ is equivalent to:
   • $(-3, \frac{5\pi}{3} - 2\pi) = (-3, -\frac{\pi}{3})$,
   • Rotating by multiples of $2\pi$, the equivalent points are:
     • $(3, \frac{2\pi}{3})$,
     • $(3, -\frac{5\pi}{3})$,
     • $(3, -\frac{4\pi}{3})$,
     • $(-3, -\frac{\pi}{3})$.
2. The point $(3, -\frac{5\pi}{3})$ is NOT the same.

Answer: (b) $(3, -\frac{5\pi}{3})$

If you need further clarification, let me know!

5 Follow-up Questions

1. Can you generalize how to convert polar coordinates to rectangular coordinates?
2. How would you solve a system of equations graphically?
3. Can you derive all equivalent polar coordinates for a given point?
4. What is the significance of cotangent in trigonometric simplifications?
5. How would you explain the relationship between rectangular and polar forms to a