Solution of triangles: Difference between revisions
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In [[trigonometry]], to '''solve a triangle''' is to find the three angles and the lengths of the three sides of the [[triangle]] when given some, but not all of that information. In particular: |
In [[trigonometry]], to '''solve a triangle''' is to find the three angles and the lengths of the three sides of the [[triangle]] when given some, but not all of that information. In particular: |
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* If one knows two of the angles one can find the third by using the fact that the sum of the three must be 180°. |
* If one knows two of the angles one can find the third by using the fact that the sum of the three must be 180°. |
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* (SSS) If one knows the lengths of the three sides, one can find the three angles by using the [[law of cosines]]. |
* '''(SSS)''' If one knows the lengths of the three sides, one can find the three angles by using the [[law of cosines]]. |
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* (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. |
* '''(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. |
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* (SSA) If one knows the lengths of two sides and an angle between one of those and the third side, one can find the third length and the other angles by using the [[law of sines]], in some cases [[up to]] a choice between two possible solutions. |
* '''(SSA)''' If one knows the lengths of two sides and an angle between one of those and the third side, one can find the third length and the other angles by using the [[law of sines]], in some cases [[up to]] a choice between two possible solutions. |
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* (SAA) If one knows the length of one side and at least two of the angles, one can find the lengths of the other sides by using the [[law of sines]]. |
* '''(SAA)''' If one knows the length of one side and at least two of the angles, one can find the lengths of the other sides by using the [[law of sines]]. |
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In some cases, the [[law of tangents]] can also be used. |
In some cases, the [[law of tangents]] can also be used. |
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Revision as of 14:13, 12 April 2009
In trigonometry, to solve a triangle is to find the three angles and the lengths of the three sides of the triangle 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 be 180°.
- (SSS) If one knows the lengths of the three sides, one can find the three angles by using the law of cosines.
- (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.
- (SSA) If one knows the lengths of two sides and an angle between one of those and the third side, one can find the third length and the other angles by using the law of sines, in some cases up to a choice between two possible solutions.
- (SAA) If one knows the length of one side and at least two of the angles, one can find the lengths of the other sides by using the law of sines.
In some cases, the law of tangents can also be used.
Mollweide's formula can be used to check solutions.