Law of sine in non-right triangle

The law of sine - you need this knowledge

• The simple trigonometric functions sin, cos and tan are only valid in a right-angled triangle because they refer to the hypotenuse and legs of this triangle.
• Nevertheless, one is not lost when calculating sides and angles in a non-right-angled triangle, because there the law of sines and (the somewhat more difficult to understand) law of cosines.
• With the law of sine, sides and the sine of the opposite (!) angle always in the same ratio.
• In formulas the sentence is a/sin α = b/sin β = c/sin γ. The angle γ is arbitrary here and not 90°.
• To calculate sides and/or angles, two matching parts of these continuous proportions are selected. In this case, the law of sine "decomposes" into three equations.


Angle calculation on a triangle - explained step by step

Don't panic about math problems! With a good sketch and the right formulas, the...

By the way, other formulations of the theorem are a/b = sin α/sin β (and each exchanged with the further angle and the third side).

Example calculation in the non-right-angled triangle

As an example, a general (i.e. non-right-angled) triangle should be chosen here, where a = 3 cm, b = 5 cm and the angle β = 50° is given (this constellation corresponds to the congruence theorem sws). You are looking for the third side c and the two angles α and γ.

1. You first calculate the angle α, because this is opposite the given side a. You set up: a/sin α = b/sin β, insert the given quantities: 3/sin α = 5/sin 50°. Now multiply this proportion "crosswise" and get: 3 _* sin 50° = 5 _* sin α and therefore sin α = 0.46 and with INV SIN (sin^-1): α = 27,4°.
2. You can easily calculate the third angle γ, because γ = 180° - 27.4° - 50° = 102.6° (sum of angles in a triangle) applies.
3. You can now also calculate the third missing side c using the law of sines. You choose (for example): b/sin β = c/sin γ and put in: 5/sin 50° = c/sin 102.6° and get from this c = 6.37 cm (the largest angle is also here the largest side opposite).

By the way: Problems in which a non-right-angled triangle has three sides (sss) or two sides and the included angles (sws) are given cannot be solved with the law of sines (but with the law of cosines, see link above).