One of the most important geometric shapes is the right triangle. At its essence, it is a triangle with a 90-degree angle (right angle)—a simple definition but has countless applications across many disciplines.
Geometry
The right triangle consists of three sides \(a\), \(b\), and \(c\). The longest side is what we call the hypotenuse \(c\), while the other two sides are its legs \(a\) and \(b\). The angle between the legs is always equal to 90 degrees, while the angle between each leg and hypotenuse, \(a-c\) or \(b-c\), can be any as long as the sum of the interior angles is always 180 degrees.
Pythagorean Theorem
One key relationship among the sides of a right triangle is the Pythagorean theorem. It states that the square of the hypotenuse is the sum of the squares of its legs. Mathematically speaking, it is:
\(c^2=a^2+b^2\)
From this theorem, it is possible to compute any side of the right triangle if we have the length of its other two sides.
Six Trigonometric Functions
Associated with the right triangle are the six trigonometric functions. These expressions relate the interior angles between a leg and the hypotenuse, say \(a-c\) or \(b-c\), with the sides of the right triangle. We do this by expressing the sides as ratios.
Before we dive into these ratios, it's essential to get some terms straight. Say we have a right triangle. Consider an interior angle \(\theta\) between one of the legs and the hypotenuse. From \(\theta\), we name the sides of the triangle the following:
- The opposite side is the leg subtended by \(\theta\).
- The adjacent side is the leg next to \(\theta\).
- The hypotenuse is the longest side of the right triangle. Furthermore, it is always opposite to the right angle.
With these terms, we can dive straight into the functions, which are:
These six functions allow us to determine the angles of a right triangle.
Summary
One of the most important geometric shapes is the right triangle. It is a triangle with a 90-degree angle (right angle).
The right triangle consists of three sides \(a\), \(b\), and \(c\). The longest side is what we call the hypotenuse \(c\), while the other two sides are its legs \(a\) and \(b\).
The Pythagorean theorem is an important relationship among the sides of a right triangle. It states that the square of the hypotenuse is the sum of the squares of its legs. Mathematically speaking, it is: \(c^2=a^2+b^2\)
Associated with the right triangle are the six trigonometric functions which link an interior angle and the ratios of the sides of the right triangle.
The six functions are sine, cosine, tangent, cosecant, secant, and cotangent.
