Right triangle

A right triangle has one interior right angle, i.e., of 90 degrees. This triangle has several interesting properties, which will be discussed in this article.

Basic description

The right triangle has already been partially described in the main article on triangles. A right triangle has one interior angle of 90 degrees. Both of the remaining interior angles must necessarily be less than 90 degrees, otherwise the sum of the interior angles would not be equal to 180 degrees. It is even true that the sum of the two remaining angles is exactly 90 degrees.

A right-angled triangle has three sides, of course, two of which are called the hypotenuse (the red side) - these are the smaller sides - and the third side is called the hypotenuse (the blue side) - this is the longest side. The hypotenuse is always opposite the point at which the right angle is.

Pythagorean Theorem

Probably the most famous mathematical theorem ever, the Pythagorean Theorem, holds in a right triangle. Pythagoras' theorem deals with the size of the sides of a triangle. The theorem states that "The area of the square above the hypotenuse of a right triangle is equal to the sum of the area of the squares above its branches". Mathematically written:

$$c^2=a^2+b^2$$

Pythagorean Theorem in Figure: The theorem is discussed in a separate article.

area of a right triangle

In a right-angled triangle, it is very easy to calculate the content because the heights are congruent to the branches. Imagine the right triangle in the first figure. How would we calculate its content without knowing any heights? We can complete the triangle to make a rectangle. The sides of the rectangle will be the branches of the triangle and the other two sides will be added to make a rectangle: We can already calculate the area of the rectangle, it is the product of the lengths of two adjacent sides, in this case the product of the red sides AC and AB. This gives us the area of the rectangle. However, the hypotenuse BC forms the diagonal of the rectangle, which bisects the rectangle, so if we divide the area of the rectangle by two, we get the area of a right triangle. So the formula would look like this:

$$S_\triangle=\frac{b\cdot c}{2},$$

where b and c are the lengths of the branches.

The goniometric functions sine and cosine

In a right triangle, the basic goniometric functions and their relationships hold. Goniometric functions such as sine, cosine (sometimes also cosine), tangent and cotangent express the ratio between the lengths of the sides in a right triangle. Consider the first triangle again, this time with the angles highlighted.

To find the ratio of the lengths of the two sides, we take the two lengths and divide them in the order shown. For example, the ratio of the lengths of the sides of b:c (read "bé to cé") is obtained by dividing 3/5 (side b is 3, side c is 5).

The sine function then works with the ratio of two sides and one angle. Given our figure, if we take the ratio of the sides b:a, we get the value of the sine function applied to the angle β. The cosine works similarly, but for the ratio b:a and the same angle β. Thus:

$$\begin{eqnarray} \sin(\beta)&=&\frac{b}{a}\\ \cos(\beta)&=&\frac{c}{a} \end{eqnarray}$$

The sine and cosine functions always work only with those angles in a right triangle that are less than 90 degrees. In this case, the angles β and γ. Within this range, the sine and cosine functions always return a number in the interval (0, 1), which helps you remember the rule that you always divide the shorter side by the longer side - to get a number just from the interval (0, 1).

Both functions work with one branch and hypotenuse at the same time, never two branches at the same time. Since you are always dividing the shorter by the longer, the hypotenuse will always be in the denominator of the fraction. If you look at the patterns above the paragraph, the denominator is always a, which is the hypotenuse of the triangle.

The last rule is that the sine works with the opposite abscissa, while the cosine works with the adjacent abscissa. What does this mean? If we have an angle β, then the adjacent branch is the branch that comes from the point B, the branch c. The opposite branch is the other branch, the branch b. Therefore, in the formula, the side b is the side of the sine and the side c is the side of the cosine.

To summarize: the sine returns the ratio of the opposite eave to the hypotenuse, the cosine the ratio of the adjacent eave to the hypotenuse. How can we take advantage of this? If we know the angles in the triangle and the length of the hypotenuse or the eaves, we can calculate the lengths of the remaining sides. Example:

For the previous triangle:

$$|AC|=b=3,\quad|AB|=c=5,\quad\beta=30{,}96^\circ$$

What is the length of the side a? We know that:

$$\sin(\beta)=\frac{b}{a}$$

We know the value of the sinus, we also know the length of the side b, we find the length of the side a. We isolate a using equivalent modifications:

$$\begin{eqnarray} \sin(\beta)&=&\frac{b}{a}\\ a\cdot\sin(\beta)&=&b\\ a&=&\frac{b}{\sin(\beta)} \end{eqnarray}$$

(We first multiplied a by the equation, and then divided the equation by the sine.) On the right side, we already have all the known values, so we add. We calculate the value of the sine on the calculator.

$$a=\frac{b}{\sin(\beta)}=\frac{3}{0{,}514}=5{,}83.$$

The page a is 5.83 long. If you calculate the sine on the calculator, make sure you have the degree mode on. This is because often the problem is that you are calculating with radians, which is a different angular measure. However, the conversion relationship applies:

$$rad=\frac{\pi}{180}\cdot deg,$$

where deg is your value in degrees.

Goniometric functions tangent and cotangent

You can easily use these two functions in a right triangle. Again, always work with angles less than 90 degrees. However, unlike sine and cosine, these functions do not work with the hypotenuse of the triangle, but only with the branches. Thus, the tangent of an angle is equal to the ratio of the opposite branch to the adjacent branch, and the cotangent is equal to the ratio of the adjacent branch to the opposite branch.

Given the previous triangle, this is true:

$$\begin{eqnarray} \tan(\beta)&=&\frac{b}{c}\\ \mbox{cotan}(\beta)&=&\frac{c}{b} \end{eqnarray}$$

Let us try to calculate the length of the side AC (side b) in the previous triangle, using the angle β. We know its magnitude from the previous example. We know that:

$$\tan(\beta)=\frac{b}{c}$$

Let's isolate the side b:

$$\begin{eqnarray} \tan(\beta)&=&\frac{b}{c}\\ c\cdot\tan(\beta)&=&b\\ b&=&c\cdot\tan(\beta) \end{eqnarray}$$

(We just multiplied the equation by c and swapped the sides.) Now we know everything we need to calculate the length of the side b. We add:

$$b=c\cdot\tan(\beta)=5\cdot\tan(30{,}96)=5\cdot0{,}6=3$$

We can already see from the figure that we got the correct solution. Just be careful again that the tangent in the formula is a value in degrees, whereas the calculator may require the angle in radians.

To illustrate, we can calculate the same side using the cotangent:

$$\begin{eqnarray} \mbox{cotan}(\beta)&=&\frac{c}{b}\\ b\cdot\mbox{cotan}(\beta)&=&c\\ b&=&\frac{c}{\mbox{cotan}(\beta)} \end{eqnarray}$$

Add the values:

$$b=\frac{5}{\mbox{cotan}(30{,}96)}=\frac{5}{1{,}666\ldots}=3$$

Again, we get the correct result.