**Trigonometry** is a part of mathematics which deals with the study of angles. The functions used in trigonometry are named **trigonometric functions**.

In this tutorial we are going to dive into the basic trigonometric functions: sine, cosine, tangent and cotangent.

For a better understanding we are going to look into a real world example. Let’s supose we have a vehicle which drives on an incline road.

If the road level rises with 3 meter (height) for every 100 meters (slope distance), then the ratio between the height of the road (*h*) and the slope distance (*s*) is a measurements of the angle of the slope with a horizontal line. This ratio (*h/s*) is function of the angle (α), named **the sine of the angle** (α).

\frac{h}{s}&=\text{sin}(\alpha)\\

h&=s \cdot \text{sin}(\alpha)

\end{split} \end{equation*} \]

For our example above the sine of the angle (α) is:

\[ \begin{equation*} \begin{split}\text{sin}(\alpha) = \frac{3}{100} = 0.03

\end{split} \end{equation*} \]

We obtain the same result if we take the whole slope distance, from point A to point B and the total height (from point C to point B:

\[ \begin{equation*} \begin{split}\text{sin}(\alpha) = \frac{12}{400} = 0.03

\end{split} \end{equation*} \]

If we take the projection of the slope distance (100 m) on the horizontal axis we’ll get a length (*l*). The ratio between the length (l) and the slope distance (s) is also a function of the angle (α). This function is called **the cosine of the angle** (α).

\frac{l}{s}&=\text{cos}(\alpha)\\

l&=s \cdot \text{cos}(\alpha)

\end{split} \end{equation*} \]

In order to calculate the value of the cosine function we need to know the length (*l*). To do this we will apply the Pythagoras’ theorem.

From this we extract the value of the length (*l*):

l^2 &= s^2 – h^2\\

l &= \sqrt{s^2 – h^2}\\

l &= \sqrt{10000 – 9}\\

l &= 99.955

\end{split} \end{equation*} \]

Now we can easily calculate the value of the cosine function:

\[ \text{cos}(\alpha) = \frac{99.955}{100} = 0.541 \]The ratio between the height (*h*) and the length (*l*) is also function of the angle (α). This function is called **the tangent of the angle** (α).

The value of the tangent function is:

\[\text{tg}(\alpha)=\frac{3}{99.955}=0.0300135\]The ratio between the length (*l*) and the height (*h*) is also function of the angle (α). This function is called **the cotangent of the angle** (α).

The value of the cotangent function is:

\[\text{ctg}(\alpha)=\frac{99.955}{3}=33.31833\]From the definitions of the tangent and cotangent functions we can observe that the cotangent function is the inverse of the tangent function:

\[\text{ctg}(\alpha)=\frac{1}{\text{tg}(\alpha)}\]Now that we’ve defined the main trigonometric function, we can write down the easiest way to remember them. For a given right triangle with one hypotenuse and two cathetus (opposite and adjacent) the definitions of the main trigonometric functions are given below.

Trigonometric functions definitions for a right triangle:

\[\text{sin}(\alpha)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{h}\] | \[\text{cos}(\alpha)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{h}\] |

\[\text{tg}(\alpha)=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}\] | \[\text{ctg}(\alpha)=\frac{\text{adjacent}}{\text{opposite}}=\frac{b}{a}\] |

There are two more trigonometric function which rarely used in mathematics and engineering. These are the **secant** and **cosecant**. These are defined as:

\[\text{sec}(\alpha)=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{h}{b}\] | \[\text{cosec}(\alpha)=\frac{\text{hypotenuse}}{\text{opposite}}=\frac{h}{a}\] |

The secant and cosecant functions are mainly used in astronomy and navigation.

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