Converting a point from polar coordinates $(r, \theta)$ to rectangular coordinates $(x, y)$ involves using specific trigonometric formulas.
To convert a point from polar form $(r, \theta)$, where $r$ is the distance from the origin and $\theta$ is the angle from the positive x-axis, to its equivalent rectangular form $(x, y)$, you use the following steps and equations:
The Conversion Process
The conversion relies on the relationship between the polar coordinates $(r, \theta)$ and the rectangular coordinates $(x, y)$, which stems from basic trigonometry applied to a right triangle formed by the point, the origin, and its projection onto the x-axis.
Step-by-Step Conversion
Here are the exact steps required for the conversion:
- Step 1: Find the x-coordinate for the rectangular coordinate form of the point by using the equation $x = r \cos \theta$.
- Step 2: Find the y-coordinate for the rectangular coordinate form of the point by using the equation $y = r \sin \theta$.
By calculating $x$ and $y$ using these formulas with the given values of $r$ and $\theta$, you obtain the rectangular coordinates $(x, y)$ that correspond to the polar coordinates $(r, \theta)$.
Understanding the Formulas
- x = r cos θ: This formula projects the distance $r$ onto the x-axis based on the angle $\theta$.
- y = r sin θ: This formula projects the distance $r$ onto the y-axis based on the angle $\theta$.
These formulas are fundamental in coordinate geometry and are derived directly from the definitions of cosine and sine in a right triangle: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$ and $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$.
Example: Converting Polar to Rectangular Coordinates
Let's convert the polar coordinate point $(r, \theta) = (4, \frac{\pi}{3})$ to rectangular coordinates $(x, y)$.
Here, $r = 4$ and $\theta = \frac{\pi}{3}$ (which is 60 degrees).
-
Find x:
$x = r \cos \theta = 4 \cos(\frac{\pi}{3})$
Since $\cos(\frac{\pi}{3}) = \frac{1}{2}$,
$x = 4 \times \frac{1}{2} = 2$. -
Find y:
$y = r \sin \theta = 4 \sin(\frac{\pi}{3})$
Since $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$,
$y = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}$.
So, the rectangular coordinates are $(x, y) = (2, 2\sqrt{3})$.
Summary of Example
| Polar Coordinates (r, θ) | Calculation for x = r cos θ | Calculation for y = r sin θ | Rectangular Coordinates (x, y) |
|---|---|---|---|
| $(4, \frac{\pi}{3})$ | $x = 4 \cos(\frac{\pi}{3}) = 4 \times \frac{1}{2} = 2$ | $y = 4 \sin(\frac{\pi}{3}) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}$ | $(2, 2\sqrt{3})$ |
This process allows you to easily translate points between the two coordinate systems, which is essential in various fields like physics, engineering, and mathematics.
