Questions: Convert the polar equation to rectangular coordinates. (Use variables x and y as needed.) θ=3 π

Transcript text: Convert the polar equation to rectangular coordinates. (Use variables $x$ and $y$ as needed.) \[ \theta=3 \pi \]

Solution

Solution Steps

To convert the polar equation $\theta = 3\pi$ to rectangular coordinates, we need to use the relationship between polar and rectangular coordinates. In polar coordinates, $\theta$ represents the angle, and in rectangular coordinates, we use $x$ and $y$. The relationship is given by: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \] Since $\theta = 3\pi$, we can substitute this value into the equations for $x$ and $y$.

Solution Approach

Use the given $\theta$ value in the polar coordinate system.
Apply the trigonometric identities to convert $\theta$ to rectangular coordinates.

Step 1: Convert $\theta$ to Rectangular Coordinates

Given the polar angle $\theta = 3\pi$, we can find the rectangular coordinates $x$ and $y$ using the relationships: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \] Assuming $r = 1$ for simplicity, we substitute $\theta$ into the equations.

Step 2: Calculate $x$ and $y$

Substituting $\theta = 3\pi$: \[ x = 1 \cdot \cos(3\pi) = -1.0 \] \[ y = 1 \cdot \sin(3\pi) = 3.6739 \times 10^{-16} \]

Step 3: Present the Results

The rectangular coordinates corresponding to the polar angle $\theta = 3\pi$ are: \[ x = -1.0 \] \[ y \approx 3.6739 \times 10^{-16} \]