Questions: The polar coordinates of a point are given. Find the rectangular coordinates of this point. (3, 5π/4) What are the rectangular coordinates of this point? (Type an ordered pair. Simplify your answer, including any radicals. Rationalize all denominators. Use integers or fractions for any numbers in the expression.)

Solution Steps

To convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\). Here, \(r = 3\) and \(\theta = \frac{5\pi}{4}\).

Given the polar coordinates \((r, \theta) = \left(3, \frac{5\pi}{4}\right)\), we can find the rectangular coordinates \((x, y)\) using the formulas: \[ x = r \cdot \cos(\theta) \quad \text{and} \quad y = r \cdot \sin(\theta) \]

Substituting the values of \(r\) and \(\theta\): \[ x = 3 \cdot \cos\left(\frac{5\pi}{4}\right) \quad \text{and} \quad y = 3 \cdot \sin\left(\frac{5\pi}{4}\right) \] Calculating these values gives: \[ x \approx -2.1213 \quad \text{and} \quad y \approx -2.1213 \]

Thus, the rectangular coordinates can be expressed as: \[ (x, y) \approx (-2.1213, -2.1213) \]

Final Answer