To convert polar coordinates $(r, \theta)$ into Cartesian coordinates $(x, y)$, we use the formulas:

$x = r \cos(\theta)$ $y = r \sin(\theta)$

1. Converting $(5, 0)$

• $r = 5$
• $\theta = 0$

Substituting in: $x = 5 \cos(0) = 5 \cdot 1 = 5$ $y = 5 \sin(0) = 5 \cdot 0 = 0$

So, $(5, 0)$ in Cartesian coordinates is: $(5, 0)$

2. Converting $(5, \pi)$

• $r = 5$
• $\theta = \pi$

Substituting in: $x = 5 \cos(\pi) = 5 \cdot (-1) = -5$ $y = 5 \sin(\pi) = 5 \cdot 0 = 0$

So, $(5, \pi)$ in Cartesian coordinates is: $(-5, 0)$

Final Answer:

• $(5, 0)$ in Cartesian coordinates is $(5, 0)$.
• $(5, \pi)$ in Cartesian coordinates is $(-5, 0)$.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do polar coordinates relate to the Cartesian plane?
2. What if $\theta$ is not a multiple of $\pi$, how would the calculations differ?
3. Can the same Cartesian point have multiple polar coordinates?
4. What is the significance of $r$ and $\theta$ in polar coordinates?
5. How are negative $r$ values interpreted in polar coordinates?

Tip: Remember that in polar coordinates, a positive $r$ with an angle $\theta + \pi$ points directly opposite to the angle $\theta$.