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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.
In many cases it is sufficient to establish the equality of three corresponding parts and use one of the following results to deduce the congruence of the two triangles. Determining congruence The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS), two angles and the side between them ...
Obtuse case. Figure 7b cuts a hexagon in two different ways into smaller pieces, yielding a proof of the law of cosines in the case that the angle γ is obtuse. We have in pink, the areas a 2, b 2, and −2ab cos γ on the left and c 2 on the right; in blue, the triangle ABC twice, on the left, as well as on the right.
Special cases where this relationship does not hold, or is ambiguous, include cases like: Three or more collinear points, where the circumcircles are of infinite radii . Four or more points on a perfect circle, where the triangulation is ambiguous and all circumcenters are trivially identical.
This is the ambiguous case and two different triangles can be formed from the given information. UPDATED JackOL31 16:22, 5 December 2009 (UTC) Yes, definitely an improvement! (I've tried myself, and given up on my attempt to make it simple, but I think your draft makes it as clear as the situation permits.)
The team was the first to take a deep dive into the genetics behind the crop’s nutritional properties—and make a case for why chia seeds may be one solution to the world’s hunger and ...
25 hostess gifts from Walmart are way better than a bottle of wine
Take a hyperbolic plane whose Gaussian curvature is .Given a hyperbolic triangle with angles ,, and side lengths =, =, and =, the following two rules hold.The first is an analogue of Euclidean law of cosines, expressing the length of one side in terms of the other two and the angle between the latter: