What is trigonometry

Trigonometry¶

In trigonometry, angle sizes in triangles are examined. These play an important role in many areas of mathematics and physics.

The sine theorem¶

Every acute triangle can be broken down into two right triangles by drawing a contour line. One denotes the point of intersection of the height with the route as , then applies to the partial triangle :

Subdivision of a triangle to prove the sine theorem.

For the partial triangle Shall apply accordingly:

If one sets the above two equations for the same, one obtains the following relation:

If you draw in all three contour lines, you get a corresponding size equation for each. If these are converted into proportional equations, the following "sine law" results:

The sine law is usually further summarized into a single equation:

The side lengths of a triangle are therefore in the same ratio to one another as the sine values ​​of the respective opposite angles.

The sine theorem also applies to obtuse triangles. It can be used, for example, to calculate missing pieces of a triangle if there are two side lengths and an opposite angle or one side length and two angles.

The Cosine Theorem¶

In every triangle, the square of one side is equal to the sum of the squares of the other two side lengths, minus the double product of these two side lengths and the cosine value of the included angle. For example, the following applies to any angle values:

Is , so is , and thus . The Pythagorean theorem is thus a special case of the cosine theorem for right triangles.

For the other two sides and Shall apply accordingly:

The cosine theorem can be used to construct triangles if either all three side lengths or two side lengths and the angle they enclose are given.

Example:

  • What are the values ​​of the angles of a triangle, its sides , and are long?

    According to the cosine theorem:

    If you insert the given values, you get:

    As expected, the following applies to the sum of the interior angles .