Sine Formula Sine Formula The law of sines (also known as the sine law, sine formula, or sine rule) is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law, a /sinA = b/sin B = c/sin B where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles (see the figure to the right). Sometimes the law is stated using the reciprocal in this equation: SinA / a = sinB / b = sin C / c The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the nonenclosed angles are known. In some such cases, the formula gives two possible values for the enclosed angle, leading to an ambiguous case.The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in a general triangle, the other being the law of cosines. The sine law can be used to prove the angle sum identity for sine when α and β are each between 0 and 90 degrees.To prove this, make an arbitrary triangle with sides a, b, and c with corresponding arbitrary angles A, B and C. Draw a perpendicular to c from angle C. This will split the angle C into two different angles, α and β, that are less than 90 degrees, where we choose to have α to be on the same side as A and β be on the same side as B. Use the sine law identity that relates side c and side a. Know More About :- Bar Graph Definition
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Notice that the perpendicular makes two right angles triangles, also note that sin(A) = cos(α), sin(B) = cos(β) and that c = a sin(β) + b sin(α). After making these substitutions you should have sin(C) =sin(α + β) = sin(β)cos(α) + (b/a)sin(α)cos(α). Now apply the sine law identity that relates sides b and a and make the substitutions noted before. Now substitute this expression for (b/a) into the original equation for sin(α + β) and you will have the angle sum identity for α and β in terms of sine. The only thing that was used in the proof that was not a definition was the sine law. Thus the sine law is equivalent to the angle sum identity when the angles sum is between 0 and 180 degrees and when each individual angle is between 0 and 90 degrees. Putting any of the four vertices in the role of O yields four such identities, but in a sense at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity. One reason to be interested in this "independence" relation is this: It is widely known that three angles are the angles of some triangle if and only if their sum is a half-circle. What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be a half-circle. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by this sines law further reduce the number of degrees of freedom, not from 8 down to 4, but only from 8 down to 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5dimensional. Ptolemy's theorem implies the theorem of Pythagoras. The latter serves as a foundation of Trigonometry, the branch of mathematics that deals with relationships between the sides and angles of a triangle. In the language of Trigonometry, Pythagorean Theorem reads sin²(A) + cos²(A) = 1, where A is one of the internal angles of a right triangle. If the hypotenuse of the triangle is of length 1, then sin(A) is the length of the side opposite to the angle A, cos(A) is the length of the adjacent side.Ptolemy's theorem also provides an elegant way to prove other trigonometric identities. In a little while, I'll prove the addition and subtraction formulas for sine:
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