**The cosine theorem** is a fundamentally important part of trigonometry and plays an important role in geometry. It is used to describe the relationship between the sides and angles of a triangle, especially in the case of a non-right-angled triangle. **The formula for the cosine theorem is: c² = a² b²- 2ab* cos(γ).**

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## Basics: What is the cosine theorem?

The cosine theorem, also known as Carnot's cosine theorem, makes it possible to calculate a side or an angle in any triangle if certain other sizes of the triangle are known. It extends the range of Pythagoras' theorem, which is only valid for right-angled triangles.

The general formula of the cosine theorem is **c² = a² b² - 2from * cos(γ)** where $a,b$ and $c$ are the side lengths of a triangle and $γ$ which is equal to the side length $c$ is the opposite angle.

## Areas of application

The cosine theorem is useful for

- Determining the length of a side in a triangle if two sides and the included angle are known.
- Calculating an angle in a triangle if all three side lengths are known.

## Calculating angles with the cosine theorem

If you know the lengths of all three sides of a triangle and want to calculate one of the angles, you can rearrange the cosine theorem as follows:

$γ$

It is important to note that

- $γ$ is the angle corresponding to the side $c$ is opposite.
- If you want to determine a different angle, you must exchange the corresponding side lengths in the formula.

## Example task

Suppose you have a triangle with the side lengths $a=5$ cm, $b=6$ cm and $c=7$ cm and want the angle $γ$ which $c$ is opposite.

Insert into the rearranged formula:

$γ$

γ = cos⁻¹ (6 / 30)

Now you can use the arc cosine (or the inverse cosine) to get $γ$ to determine it.

## Conclusion

The cosine theorem provides a powerful tool for solving triangle problems where the Pythagorean theorem is not applicable. Its application can effectively determine both angles and sides in non-rectangular triangles.