Math Problem Statement

Convert θ = 3π/4 to rectangular form.

Solution

To convert θ=3π4\theta = \frac{3\pi}{4} to rectangular form, we should think about the relationship between polar and rectangular coordinates.

For a line at an angle θ=3π4\theta = \frac{3\pi}{4} with the origin in polar coordinates, this angle corresponds to a line where the slope is determined by tan(3π4)\tan\left(\frac{3\pi}{4}\right).

  1. Calculate the Slope: tan(3π4)=1\tan\left(\frac{3\pi}{4}\right) = -1 Therefore, the line has a slope of 1-1.

  2. Equation of the Line: In rectangular form, this is the line y=xy = -x, since the angle 3π4\frac{3\pi}{4} in polar coordinates corresponds to the line where yy and xx have opposite signs with equal magnitudes (slope of 1-1).

Answer:

The correct answer is: D. y=xy = -x

Would you like further details or have any questions?


Here are five related questions for deeper understanding:

  1. How do you convert from polar to rectangular coordinates in general?
  2. What are some common angles and their corresponding slopes in rectangular form?
  3. Why does tan(3π4)\tan\left(\frac{3\pi}{4}\right) equal 1-1?
  4. How would you find the rectangular form of a polar equation with a radius, such as r=2r = 2?
  5. What is the rectangular form of θ=π4\theta = \frac{\pi}{4}?

Tip: When converting polar angles to rectangular equations, think about the trigonometric values of the angle, as they often determine the slope or relationship between xx and yy.

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Math Problem Analysis

Mathematical Concepts

Polar and Rectangular Coordinates
Trigonometry
Coordinate Conversion

Formulas

tan(θ) = y/x
Slope formula: m = tan(θ)

Theorems

Definition of Polar to Rectangular Conversion

Suitable Grade Level

Grades 10-12