Math Problem Statement

Convert the polar coordinates (r, φ) with r = 5 and φ = 3π/4 to Cartesian form.

Solution

The problem is asking for the conversion of polar coordinates (r,φ)(r, \varphi) to Cartesian coordinates.

Given:

  • r=5r = 5
  • φ=3π4\varphi = \frac{3\pi}{4}

To convert from polar to Cartesian coordinates, we use the following formulas: x=rcos(φ)x = r \cos(\varphi) y=rsin(φ)y = r \sin(\varphi)

Step-by-Step Solution:

  1. Calculate xx: x=5cos(3π4)=522=522x = 5 \cos\left(\frac{3\pi}{4}\right) = 5 \cdot -\frac{\sqrt{2}}{2} = -\frac{5\sqrt{2}}{2}

  2. Calculate yy: y=5sin(3π4)=522=522y = 5 \sin\left(\frac{3\pi}{4}\right) = 5 \cdot \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}

Cartesian Coordinates

The Cartesian form of the point is: z=x+yi=522+522iz = x + yi = -\frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}i

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions for further practice:

  1. How would you convert from Cartesian to polar coordinates?
  2. What are the polar coordinates of a point in the third quadrant with the same rr and angle φ\varphi?
  3. How do trigonometric functions behave for different quadrants?
  4. How would this change if r=5r = -5?
  5. Can you find the modulus and argument of the complex number in Cartesian form?

Tip: Always check which quadrant your angle falls into to determine the correct signs for xx and yy in Cartesian coordinates.

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Math Problem Analysis

Mathematical Concepts

Polar to Cartesian Coordinate Conversion
Trigonometry

Formulas

x = r * cos(φ)
y = r * sin(φ)

Theorems

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Suitable Grade Level

Grades 10-12