**The sine theorem** is one of the most important theorems in trigonometry for determining sides and angles in a triangle. It is particularly important for non-right-angled triangles and establishes a connection between the side lengths and the angles of a triangle. The sine theorem is $).$

## Calculator

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- Enter the radians of the opposite cathetus and hypotenuse into the calculator.
- Alternatively, you can also enter the angle degrees.
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## Basic principle of the sine theorem

Given a triangle ABC with the side lengths a, b and c and the opposite angles α, β and γ.

The sine theorem reads:

This means that the ratio of a side length to the sine of the opposite angle is the same for all three pairs of sides and angles.

**Applications of the sine theorem**

**Determining a missing side length:**If two sides and an angle of a triangle are given, the third side length can be calculated by rearranging the sine theorem.**Determining a missing angle:**If all three side lengths of a triangle are given, a missing angle can be calculated using the sine theorem.**Checking the existence of a triangle:**The sine theorem can also be used to check whether a triangle with given side lengths and angles can exist at all.

## Advantages of the sine theorem

- It can be used universally for all triangles, not just right-angled triangles.
- Enables simple calculations even for complex triangle problems.
- Many geometric problems can be solved in combination with other trigonometric theorems.

## Application example

Suppose we have a triangle ABC with given side lengths a = 7 cm and b = 9 cm and an angle α = 45°. How large is the angle β?

Using the sine theorem:

$a / sin(α) = b /sin(β) $

If we insert the values and solve for β, we get

$sin(β)= (b xsin(α)) / a $

By inserting the given values and calculating β with the arc sine, you obtain the value of the angle β.

## Conclusion

The sine theorem is a powerful tool in trigonometry that enables efficient determination of side lengths and angles in a triangle. Through its understanding and application, many problems in geometry can be solved. It is essential to have the sine theorem in your repertoire alongside other important trigonometric theorems.