Solution of triangles

# Solution of triangles

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In trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, to solve a triangle is to find the three angles and the lengths of the three sides of the triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

when given some, but not all of that information. In particular:
• If one knows two of the angles one can find the third by using the fact that the sum of the three must always be 180°.
• (SSS) If one knows the lengths of the three sides, one can find the first angle by using the law of cosines
Law of cosines
In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...

, and proceed then with the much easier law of sines
Law of sines
In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...

.
• (SAS) If one knows the lengths of two of the sides and the angle between them, one can find the length of the third side by using the law of cosines
Law of cosines
In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...

, and proceed then with the much easier law of sines
Law of sines
In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...

.
• (SSA) If one knows the lengths of two sides and an angle between one of those and the third side, one can find the second angle by using the law of sines
Law of sines
In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...

, and then proceed by using the sum of angles within a triangle to find the third angle, finally using the law of sines
Law of sines
In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...

again for finding the remaining side lengths, in some cases up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

a choice between two possible solutions.
• (AAS) & (ASA) If one knows the length of one side and at least two of the angles, one can find the third angle by using the sum of angles within a triangle, and the lengths of the other two sides by using the law of sines
Law of sines
In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...

.

To check the solutions, Mollweide's formula
Mollweide's formula
In trigonometry, Mollweide's formula, sometimes referred to in older texts as Mollweide's equations, named after Karl Mollweide, is a set of two relationships between sides and angles in a triangle...

can be used. In some cases, the law of tangents
Law of tangents
In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposite sides....

may also be useful.

When solving spherical triangles, the half-side formula
Half-side formula
In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles....

e are used.