Basic properties of trigonometric functions

For a right triangle we can establish certain relationships between the trigonometric functions, that are valid for any angle (α).

Trigonometric functions properties:

\[\text{sin}^2(\alpha)+\text{cos}^2(\alpha)=1\]	The sum between the sine at the power of two and the cosine at the power of two will be always 1 for any angle (α)
\[\text{tg}(\alpha) \cdot \text{ctg}(\alpha)=1\]	The product between the tangent and cotangent of an angle (α) is always 1
\[\text{sec}(\alpha) = \frac{1}{\text{cos}(\alpha)}\]	The secant function is the ratio between 1 and the cosine function for any angle (α)
\[\text{cosec}(\alpha) = \frac{1}{\text{sin}(\alpha)}\]	The cosecant function is the ratio between 1 and the sine function for any angle (α)
\[\text{tg}(\alpha) = \frac{\text{sin}(\alpha)}{\text{cos}(\alpha)}=\frac{1}{\text{ctg}(\alpha)}\]	The tangent function is the ratio between 1 and the cotangent function for any angle (α)
\[\text{ctg}(\alpha) = \frac{\text{cos}(\alpha)}{\text{sin}(\alpha)}=\frac{1}{\text{tg}(\alpha)}\]	The cotangent function is the ratio between 1 and the tangent function for any angle (α)

Further we are going to look over the values of the trigonometric functions for some basic angles: 30°, 45° and 60°.

First we’ll take a square with the length of the side of 1. The unit of the length of the side is irrelevant for this example.

If we draw a diagonal from point A to point C, we’ll get a right triangle (ADC). The hypotenuse has the length of square root of 2. We find this by applying Pythagoras’ theorem:

\[ \begin{equation*} \begin{split}
AC^2 &= AD^2 + CD^2\\
AC^2 &= 1^2 + 1^2\\
AC^2 &= 2\\
AC &= \sqrt{2}
\end{split} \end{equation*} \]

We’ll use the square to find what are the values of the basic trigonometric function for the angle of 45°.

\[ \begin{equation*} \begin{split}
\text{sin}(45^\circ)&=\frac{\text{opposite}}{\text{hypotenuse}}&=\frac{CD}{AC}=\frac{1}{\sqrt{2}}\\
\text{cos}(45^\circ)&=\frac{\text{adjacent}}{\text{hypotenuse}}&=\frac{AD}{AC}=\frac{1}{\sqrt{2}}\\
\text{tg}(45^\circ)&=\frac{\text{opposite}}{\text{adjacent}}&=\frac{CD}{AD}=1\\
\text{ctg}(45^\circ)&=\frac{\text{adjacent}}{\text{opposite}}&=\frac{AD}{CD}=1
\end{split} \end{equation*} \]

For the angles of 30° and 60°, we’ll use an equilateral triangle with the side of length 1. The same, for this particular example the unit of the side length is irrelevant.

By drawing the height of the triangle, the segment BD, we get two equal segments, AD and DC which have the length of 1/2.

Using Pythagoras’ theorem we can calculate the height BD of the triangle:

\[ \begin{equation*} \begin{split}
AB^2 &= BD^2 + AD^2\\
BD^2 &= AB^2 – AD^2\\
BD^2 &= 1 – \frac{1}{4}\\
BD^2 &= \frac{3}{4}\\
BD &= \frac{\sqrt{3}}{2}
\end{split} \end{equation*} \]

Now we have two right triangles, ADB and CDB. We’ll use the right triangle ADB to find the values of the trigonometric functions for an angle of 60°:

\[ \begin{equation*} \begin{split}
\text{sin}(60^\circ)&=\frac{\text{opposite}}{\text{hypotenuse}}&=\frac{BD}{AB}=\frac{\sqrt{3}}{2}\\
\text{cos}(60^\circ)&=\frac{\text{adjacent}}{\text{hypotenuse}}&=\frac{AD}{AB}=\frac{1}{2}\\
\text{tg}(60^\circ)&=\frac{\text{opposite}}{\text{adjacent}}&=\frac{BD}{AD}=\sqrt{3}\\
\text{ctg}(60^\circ)&=\frac{\text{adjacent}}{\text{opposite}}&=\frac{AD}{CD}=\frac{1}{\sqrt{3}}
\end{split} \end{equation*} \]

To find the values of the trigonometric function for the angle of 30°, we’ll use the triangle BDC:

\[ \begin{equation*} \begin{split}
\text{sin}(30^\circ)&=\frac{\text{opposite}}{\text{hypotenuse}}&=\frac{CD}{BC}=\frac{1}{2}\\
\text{cos}(30^\circ)&=\frac{\text{adjacent}}{\text{hypotenuse}}&=\frac{BD}{BC}=\frac{\sqrt{3}}{2}\\
\text{tg}(30^\circ)&=\frac{\text{opposite}}{\text{adjacent}}&=\frac{CD}{BD}=\frac{1}{\sqrt{3}}\\
\text{ctg}(30^\circ)&=\frac{\text{adjacent}}{\text{opposite}}&=\frac{BD}{CD}=\sqrt{3}
\end{split} \end{equation*} \]

The angles of 30°, 45° and 60° are very common in geometry so knowing how to calculate the trigonometric functions for them it’s quite important.

As an exercise you can use the values of the trigonometric functions for the angles 30°, 45° and 60° to demonstrate the properties stated at the beginning of the tutorial.

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