**x**is the x-coordinate of the point in Cartesian coordinates.**y**is the y-coordinate of the point in Cartesian coordinates.**r**is the radial coordinate of the point in polar coordinates.**θ**is the angular coordinate of the point in polar coordinates.

**(r, θ)**and Cartesian coordinates

**(x, y)**:

## How to Convert from Polar to Cartesian Coordinates

Converting from the polar to the Cartesian coordinates of a point is easy.## Question

What is a point described by the polar coordinates (8, 30°) in Cartesian coordinates?## Step-by-Step:

## Find the X-Coordinate

## 2

Find r and θ from the polar coordinates given in the question.
In our example, the polar coordinates of the point is (8, 30°). They are represented in the formula by (r, θ).

(r, θ) = (8, 30°) ∴ r = 8, θ = 30°

## 3

Substitute r and θ into the formula.
The

x-coordinate = 8 cos (30°)

x-coordinate = 8 × 0.87

x-coordinate = 6.9

**x-coordinate**is**6.9**## Find the Y-Coordinate

## 4

## 5

Substitute r and θ into the formula.
The

y-coordinate = 8 sin (30°)

y-coordinate = 8 × 0.5

y-coordinate = 4

**y-coordinate**is**4**## 6

Write down the Cartesian coordinates as a pair of numbers in brackets, separated by a comma.
The x-coordinate (

**6.9**) found in**Step 3**goes on the left. The y-coordinate (**4**) found in**Step 5**goes on the right.## Answer:

The polar coordinates (8, 30°) become (6.9, 4) when converted to Cartesian coordinates.## Interactive Widget

Here is an interactive widget to help you learn about converting between Cartesian and polar coordinates.## Why Do the Formulas Work?

Polar coordinates form a right triangle: The radial coordinate is the hypotenuse and the angular coordinate is the angle.-
The x-coordinate is the adjacent of the triangle
When the hypotenuse and angle are known, use the cosine to find the adjacent:
x = r cos θ
- The y-coordinate is the opposite of the triangle.
When the hypotenuse and angle are known, use the sine to find the opposite:
y = r sin θ
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## Worksheet

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