Law of cosines
In trigonometry, the cosine law establishes a relationship between the three sides of a triangle and the cosine of the three angles of the triangle.
- 2.1 Numerical Example
- 2.2 congruence
Formulation
For the three sides, and a triangle and the angle opposite the side (i.e., the angle between the sides and past ) the following applies:
Correspondingly apply to the other angles:
The Pythagorean theorem as a special case of the cosine law
With so in a right-angled triangle, is considered. This arises as a special case of the cosine law in the right-angled triangle the Pythagorean theorem:
Therefore, the cosine rule represents a generalization of the Pythagorean theorem, also called extended theorem of Pythagoras.
Applications
Numerical example
The following numerical values are rough approximations. In a triangle ABC the following page sizes are known ( designations as usual):
Wanted is the angular size ( designations as usual).
Congruence
The congruence SSS ( side-side - side) and SWS ( side - angle - side) state that a triangle is completely determined by specifying three sides or two sides and their intermediate angle. Alternatively, you can also specify any two vectors from which the included angle can be calculated. The law of cosines it possible in these cases from the three pieces of a given fourth piece, namely, an angle ( in the case of SSS) or the third side (in the case SWS) to be calculated. If you want to then determine the remaining angles of a triangle, you can apply the law of cosines (with adjusted to the angle desired page numbers ) or the law of sines optional again. The last angle is calculated most conveniently via the angle sum of 180 °.
When only one side and two angles are given ( congruence SWW or WSW) or two sides and the opposite angle of the larger side ( congruence SSW), one first computes one of the missing pieces with the law of sines and the missing angle on the angle sum before can determine the third side with the law of cosines.
Evidence
The following evidence is required. For the proof must be modified slightly. For the law of cosines yields directly from the theorem of Pythagoras.
In the sub- triangles of the Pythagorean theorem to be applied to find an arithmetic expression for. This requires the squares of the Kathetenlängen this part of the triangle:
After Pythagoras applies for the left part of the triangle:
Thus, the two arithmetic expressions found above are added:
Now applies
With the conclusion
Substituting this intermediate result into the equation for the assertion follows:
Generalization
With vectors in real Skalarprodukträumen, ie vector spaces with scalar product, and the law of cosines can be easily generalized. Refers to
The Skalarproduktnorm, ie the length of a vector and with
The angle between the two vectors, then, for the norm of the vector: