If you have the distance between one point and another, you have a magnitude. If you have a reference angle to that point, you also have a direction. Put the magnitude together with the direction and you have a vector. Finding the coordinates of the second point is a relatively simple matter of vector analysis. Some students learn how to do this in trigonometry classes without ever realizing they are performing vector calculations. In order to perform the calculations, however, you must know where you are starting.

### Step 1

Write down your known quantities. This should include your angle, distance and reference coordinates. For example:

Angle: 37 degrees Distance: 27 feet Starting point: x0 = 3; y0 = 5

### Step 2

Calculate the cosine of the angle and multiply this by the distance.

cos(37) = 0.7986 27 * 0.7986 = 21.563

### Step 3

Calculate the sine of the angle and multiply this by the distance.

sin(37) = 0.6018 27 * 0.6018 = 16.249

### Step 4

Add the cosine result to the x-coordinate from the starting point and add the sine result to the y-coordinate to get the coordinates for the second point.

x1 = x0 + 21.563 = 3 + 21.563 = 24.563 y1 = y0 + 16.249 = 5 + 16.249 = 21.249

#### Things You Will Need

- Pen or pencil
- Scientific calculator

#### Tips

- You can check your calculations using the Pythagorean theorem. The square cosine result plus the square of the sine result should equal the square of the distance value, within a given round-off error.
- (21.563)^2 + (16.249)^2 = (27)^2
- 464.963 + 264.030 = 728.993
- 728.993 = 729