Solution by Laws of CosinesSuppose you know two sides of a triangle andthe angle between the two sides. You cannot solvethis triangle by the law of sines, since you do notknow the length of the side opposite the knownangle or the size of an angle opposite one of theknown sides. In a case of this kind you must beginby solving for the third side by applying the lawof cosines. The law of cosines is explained andproved in chapter 5 of NAVPERS 10071-B. If youare solving for a side on the basis of two knownsides and the known included angle, the law ofcosines states that in any triangle the square ofone side is equal to the sum of the squares of theother two sides minus twice the product of thesetwo sides multiplied by the cosine of the anglebetween them. This statement may be expressedin formula form as follows:For the triangle shown in figure 1-25, youknow that side c measures 10.01 ft; side b,12.00 ft; and angle A (included between them),41°24'. The cosine of 41°24' is 0.75011. Thesolution for side a is as follows:Figure 1-25.-Oblique triangle (law of cosines).Knowing the length of this side, you can nowsolve for the remaining values by applying the lawof sines.If you know all three sides of a triangle, butnone of the angles, you can determine the size ofany angle by the law of cosines, using the follow-ing formulas:For the triangle shown in figure 1-26, youknow all three sides but none of the angles. Thesolution for angle A is as follows:The angle with cosine 0.75008 measures (to thenearest minute) 41°24.Solution by Law of TangentsThe law of tangents is expressed in words asfollows: In any triangle the difference between twosides is to their sum as the tangent of half thedifference of the opposite angles is to the tangentof half their sum.1-22