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In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, = = =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.
The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. [5] Given that a triangle A B C {\displaystyle ABC} with sides a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} , and angles opposite those sides α {\displaystyle \alpha } , β {\displaystyle \beta ...
Case 3: two sides and an opposite angle given (SSA). The sine rule gives C and then we have Case 7. There are either one or two solutions. Case 4: two angles and an included side given (ASA). The four-part cotangent formulae for sets (cBaC) and (BaCb) give c and b, then A follows from the sine rule. Case 5: two angles and an opposite side given ...
This allows the two congruent purple-outline triangles and to be constructed, each with hypotenuse and angle at their base. The sum of the heights of the red and blue triangles is sin θ + sin φ {\displaystyle \sin \theta +\sin \varphi } , and this is equal to twice the height of one purple triangle, i.e. 2 sin p cos q ...
With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. [33] These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number , except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°).
Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
the third side of a triangle if two sides and an angle opposite to one of them is known (this side can also be found by two applications of the law of sines): [a] = . These formulas produce high round-off errors in floating point calculations if the triangle is very acute, i.e., if c is small relative to a and b or γ is small compared to 1.