Convert the rectangular coordinates (-4, 0) into polar form. Express the angle usin...
Check the final answer first, then review the worked steps.
Problem
Convert the rectangular coordinates (-4, 0) into polar form. Express the angle using radians in terms of pi over the interval 0 <= theta < 2*pi, with a positive value of r.
Answer
\(4(\cos(\pi) + i\sin(\pi))\)
Step-by-step solution
- Identify the rectangular coordinates: The given rectangular coordinates are $(x, y) = (-4, 0)$.
- Calculate the radial distance (r): The formula for the radial distance $r$ in polar coordinates is $r = \sqrt{x^2 + y^2}$. Substituting the given values, we get $r = \sqrt{(-4)^2 + (0)^2} = \sqrt{16 + 0} = \sqrt{16} = 4$. Since we need a positive value of $r$, we take $r=4$.
- Calculate the angle (θ): The angle $\theta$ can be found using the formula $\tan(\theta) = \frac{y}{x}$. In this case, $\tan(\theta) = \frac{0}{-4} = 0$.
- Determine the quadrant and principal angle: The point $(-4, 0)$ lies on the negative x-axis. For points on the axes, the arctangent function alone is not sufficient to determine the angle uniquely. We need to consider the quadrant. Since $x$ is negative and $y$ is zero, the point lies on the negative x-axis. The angle corresponding to the negative x-axis is $\pi$ radians.
- Verify the angle within the given interval: The problem requires the angle $\theta$ to be in the interval $0 \leq \theta < 2\pi$. Our calculated angle $\theta = \pi$ falls within this interval.
- Express in polar form: The polar form of a coordinate is given by $(r, \theta)$ or $r(\cos(\theta) + i\sin(\theta))$. Using the calculated values, the polar form is $(4, \pi)$ or $4(\cos(\pi) + i\sin(\pi))$.