Right Triangles and Vectors

A right triangle has one angle equal to 90º, or a right angle. The hypotenuse is the side opposite the ‘right’ angle, and it will always be the longest side of a right triangle.

The two remaining angles inside the triangle are less than 90º, and adding the two together always equals 90º.

If we label one of the angles less than 90º as θ, the sides of the triangle may be labeled relative to θ. There is the side opposite θ, and the side that extends from θ called the adjacent side.

Right Triangles and the Pythagorean Theorem

The Pythagorean Theorem applies to right triangles: the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two shorter sides. 

Opposite² + Adjacent² = Hypotenuse²

^{NOTE: to save space and also make the text less cumbersome to read (and write), we will use the following conventions:}

• • opposite = length of the side opposite the designated angle of the right triangle
  • adjacent = length of the side adjacent to the designated angle of the right triangle
  • hypotenuse = length of the longest side of the right triangle

Right Triangles and Trig Functions

When analyzing variables associated with right triangles, the ratio, H/L, keeps popping up again and again. For instance, consider the large right triangle with hypotenuse 2L and vertical side length 2H.  Insert a similar right triangle a vertical side length of H. This means that its hypotenuse is only L.  Both triangles have the same angle of inclination, θ.  Both H/L and θ indicate the pitch or slope of the ramp.

When a quantity keeps appearing in a calculation’s final form, mathematicians, scientists, and engineers pay attention. They gave the quantity its own name.  H/L is called “sine of θ.”

sine(θ) = sin(θ) = H / L = opposite / hypotenuse

The base of the triangle, W, is also significant in other physical phenomena.  W is the side adjacent to θ. The ratio W/L is the cosine of θ:

cosine(θ) = cos(θ) = W / L = adjacent / hypotenuse

In addition, there is a function that involves both the sin and cos functions.  The tangent of θ is defined as:

tangent(θ) = tan(θ) = H / W = opposite / adjacent = sin(θ) / cos(θ)

Calculating Vector Components from Magnitude & Direction

Since the axes that define the direction of a vector are perpendicular to each other, a right triangle also describes a vector relative to the axes.  This spatial relationship means that we use trig functions to find the axial components of vectors if we know the magnitude and direction. First, identify the angle to an axis of interest and then apply the appropriate trig function. In all cases, the hypotenuse will be the magnitude of the vector.

To convert magnitude and direction to the axial components, use trig equations.  The vector drawn on the right is 60º clockwise from the y-axis (which is north when using the convention for navigation). This means the side opposite 60º is parallel to the x-axis (East/West), and the adjacent side is parallel to the y-axis (North/South).

To find the component in the East/West direction (or X):

Opposite = X component = Magnitude * sin(Direction) = 5.0 * sin(60º) = 4.3 East

and to calculate the North/South (or Y) component:

Adjacent = Y component = Magnitude * cos(Direction) = 5.0 * cos(60º) = 2.5 North

Inverse Trig Functions

At every step in learning mathematical operations, we learn how to “undo” it at the same time. Addition and subtraction were taught together since each “undoes” the other. Now we need to learn how to undo the trig functions so we can calculate the angle if we know the ratio of the sides of a right triangle.

For the trig functions, the inverse functions are identified with a “-1” superscript or an “a” precedes the function.

Inverse sin(θ) = asin(θ) = sin¯¹(θ)

This means that asin(sin(θ)) = θ  and asin(Opposite / Hypotenuse) = θ.

Similarly,

acos(Adjacent / Hypotenuse) = θ

and

atan(Opposite / Adjacent) = θ.

On your calculator, this is sin¯¹, inv sin, arcsin, or asin.  The inverse cosine is cos¯¹, inv cos, arccos, or acos, and inverse tangent is tan¯¹, inv tan, arctan, or atan.

Calculating Magnitude & Direction from Vector Components

If we know the vector components, we use the Pythagorean Theorem to calculate the vector’s magnitude.

Magnitude = √(5.0² + 6.0²) = 7.8

Use the inverse trig functions to calculate the vector’s direction. When using the navigation convention (0º is north or in the positive y-direction) and direction increases clockwise), the vector’s direction is 50.2º:

Direction = tan¯¹(opposite / adjacent) = tan¯¹(6.0 / 5.0) = 50.2º

Using the trigonmetry convention (0º is in the positive x direction or East in the diagram) and direction increases counterclockwise), the vector’s direction is 39.8º:

Direction = tan¯¹(opposite / adjacent) = tan¯¹(5.0 / 6.0) = 39.8º

Solution converting directional components in navigation conventions to magnitude and direction.

Interactively Explore Vectors

Use the Vectors web app to visualize, conceptualize, estimate, and calculate vectors in either navigation and trigonometry direction conventions.  Many of the examples above came from the Vectors web app.

The Vectors web app is a combined simulation and practice app.  A simulation since the vector responds to a changed variable.  And a practice app since there are unlimited problems that provide visual, written, numerical, and mathematical feedback based on your answers.

Click the button to run the web app.