In the study of vectors and motion, **trigonometry** provides the framework to calculate angles and distances accurately. Once you combine vectors using vector addition, the next step often involves determining the direction or angle of the resultant vector. This is where trigonometry comes into play.
With the example exercise, the person’s northward movement and the ship’s eastward movement form a right triangle. Here, we can use trigonometric ratios such as tangent, sine, or cosine to find the relationship between the angles and the sides of the triangle.
The tangent function is especially handy when you have a right triangle and need to find the angle of the resultant vector. It is defined as the ratio between the opposite side and the adjacent side in a right triangle:
- tan(𝜃) = opposite/adjacent
In our example, this means using the speeds to find the angle:
- tan(𝜃) = 3.8 m/s (north) / 12 m/s (east)
Use the inverse tangent function (tan
-1) to compute the angle 𝜃, which gives the direction relative to the water.
This approach is fundamental in physics and engineering, where precise direction and magnitude calculations are often needed.