determine the length of side b. You could do thisas previously described by applyingHowever, the fact that side b is larger thanside a means that tan B is larger than 1 (you recallthat any angle larger than 45° has a tangent largerthan 1).You know that the cotangent is the reciprocalfunction of the tangent. Therefore, ifit follows thatA table of natural functions tells you that cot53°08' = 0.74991. Therefore,Acute Angle of RightTriangle by Sine or CosineIf you know the length of the hypotenuse andlength of a side of a right triangle, you candetermine the size of one of the acute angles byapplying the sine or the cosine of the angle.Suppose that for the triangle shown in figure 1-23,you know that the hypotenuse, c, is 5.00 ft longand that the length of side a is 3.00 ft long. Youwant to determine the size of angle A. Side a isopposite angle A; therefore,A table of natural functions tells you that an anglewith sine 0.6 measures (to the nearest minute)36°52'.Suppose that, instead of knowing the lengthof a, you know the length of b (4.00 ft). Side bis the side adjacent to angle A. You know thatA table of natural functions tells you that an anglewith cosine 0.8 measures 36°52'.If you know the size of one of the acuteangles in a right triangle and the length ofthe side opposite, you can determine thelength of the hypotenuse from the sine of theangle. Suppose that for the triangle shown infigure 1-23, you know that angle A = 36°52' andside a = 3.00 ft.If you know the size of one of the acuteangles in a right triangle and the length of the sideadjacent, you can determine the length of thehypotenuse from the cosine of the angle. Supposethat for the triangle in figure 1-23, you know thatangle A = 36°52' and side b = 4.00 ft.Tables show that cos 36°52' = 0.80003. There-fore,Solution by Law of SinesFor any triangle (right or oblique), when youknow the lengths of two sides and the size of theangle opposite one of them, or the sizes of twoangles and the length of the side opposite one ofthem, you can solve the triangle by applying thelaw of sines. The law of sines (which is explainedand proved in NAVPERS 10071-B, chapter 5)states that the lengths of the sides of any triangleare proportional to the sines of their oppositeangles. It is expressed in formula form as follows:In the triangle shown in figure 1-24,LA = 41°24', a = 8.00 ft, and b = 12.00 ft. Ifit follows thatFigure 1-24.—Oblique triangle (law of sines).1-21
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