Search results
Results from the WOW.Com Content Network
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.
Assume that two sides b, c and the angle β are known. The equation for the angle γ can be implied from the law of sines: [5] = . We denote further D = c / b sin β (the equation's right side). There are four possible cases:
There are two known points A, B, and two unknown points P 1, P 2. From P 1 and P 2 an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of P 1 and P 2. See figure; the angles measured are (α 1, β 1, α 2, β 2).
The sine rule gives b and then we have Case 7 (rotated). There are either one or two solutions. Case 6: three angles given (AAA). The supplemental cosine rule may be used to give the sides a, b, and c but, to avoid ambiguities, the half-side formulae are preferred. Case 7: two angles and two opposite sides given (SSAA). Use Napier's analogies ...
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. This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
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°).
As corollaries (multiplying or dividing the above formula in terms of and ) we obtain two dual statements to Mollweide's formulas. The first expresses the area in terms of two sides and the included angle, and the other is the law of sines: = ,
In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...