**Tangent formula**:

tan(α) = opposite cathetus / anathema

**Sine** formula:

sin(α) = opposite cathetus / hypotenuse

**Cosine** formula:

cos(α) = antecedent / hypotenuse

## Trigonometric functions for calculating angles

Depending on which lengths are known in the triangle, either the formula for the **sine**, the **cosine** or the **tangent** should be used.

## Tangent (tan) - tangent theorem

The **tangent (tan)** is calculated by dividing the **opposite cathetus** by the **anathema**.

tan(α) = opposite cathetus / anathema

### Example:

Let's start with the tangent using an example. Let's assume that our eye forms a unit with the ground and we are looking at the top of Cologne Cathedral from a distance of 100 meters. The height of Cologne Cathedral is known and is 157.38 meters. We ask ourselves at what angle is the top of Cologne Cathedral seen?

The answer can already be calculated from the available data using the tangent angle function. The tangent is calculated by dividing the opposite cathetus (height of Cologne Cathedral) by the anathema (distance to Cologne Cathedral), i.e. 157.38 meters divided by 100 meters. The result (1.5738) is a dimensionless number and is entered into the calculator. Then first press the "Shift" or "Arrow up button" button and then press the tangent function (tan).

The result then shows the number 57.57 rounded to two decimal places. And this is the angle at which we can already see Cologne Cathedral in our example, i.e. at an angle of 57.57 degrees.

## Sine (sin) - sine theorem

The **sine (sin)** is calculated by dividing the **opposite cathetus** by the **hypotenuse**.

sin(α) = opposite cathetus / hypotenuse

### Example:

Let's now move on to the sine function, which can be calculated using an analogous procedure. However, although we know the height of Cologne Cathedral in this right-angled triangle, we do not know the direct distance to Cologne Cathedral on the ground, but the direct distance between the eye and the top of Cologne Cathedral. This is also referred to as the hypotenuse in the right-angled triangle sketched here. Let's calculate the angle again from the height of Cologne Cathedral and the hypotenuse of 186.37 meters.

The value of the hypotenuse was calculated so that it corresponds to a distance to Cologne Cathedral of 100 meters. So if only the length of the distance between the eye and the tip of Cologne Cathedral (hypotenuse) and the height of Cologne Cathedral were known, and we were asking for the angle again, the sine would now be used. The angle formula for the sine is calculated by dividing the height of Cologne Cathedral by the distance between the eye and the spire of Cologne Cathedral in our example. In other words, 157.38 meters divided by 186.37 meters.

The dimensionless number of rounded 0.84 is again entered into the calculator, the "Shift" or "Arrow up key" is pressed followed by the "sin" key and the result is again the angle of around 57.6 degrees that we already know.

## Cosine (cos) - cosine theorem

The **cosine (cos)** is calculated by dividing the **anathema** by the **hypotenuse**.

cos(α) = anathema/hypotenuse

### Example:

In our example, the cosine requires the distance to Cologne Cathedral of 100 meters (ankathete) divided by the already known hypotenuse (186.37 meters). This time, dividing 100 meters by 186.37 meters results in the dimensionless number of 0.537. Typed into the calculator and again using the "Shift" key, using the "Cos" key (for cosine) gives the angle of around 57.6 degrees.