What is trigonometry

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).

\[ \begin{equation*} \begin{split}
\frac{h}{s}&=\text{sin}(\alpha)\\
h&=s \cdot \text{sin}(\alpha)
\end{split} \end{equation*} \]

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).

\[ \begin{equation*} \begin{split}
\frac{l}{s}&=\text{cos}(\alpha)\\
l&=s \cdot \text{cos}(\alpha)
\end{split} \end{equation*} \]

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.

\[s^2 = l^2 + h^2\]

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

\[ \begin{equation*} \begin{split}
l^2 &= s^2 – h^2\\
l &= \sqrt{s^2 – h^2}\\
l &= \sqrt{10000 – 9}\\
l &= 99.955
\end{split} \end{equation*} \]

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).

\[\frac{h}{l}=\text{tg}(\alpha)\]

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).

\[\frac{l}{h}=\text{ctg}(\alpha)\]

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.

For any questions or observations regarding this tutorial please use the comment form below.

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