Acute Angle of Right Triangle by Sine or Cosine

METHODS OF SOLVING TRIANGLES

Solution by Laws of Cosines

Engineering Aid 3 - Beginning Structural engineering guide book
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determine the length of side b. You could do this as previously described by applying However, the fact that side b is larger than side a means that tan B is larger than 1 (you recall that any angle larger than 45° has a tangent larger than 1). You know that the cotangent is the reciprocal function of the tangent. Therefore, if it follows that A table of natural functions tells you that cot 53°08' = 0.74991. Therefore, Acute Angle of Right Triangle by Sine or Cosine If you know the length of the hypotenuse and length of a side of a right triangle, you can determine the size of one of the acute angles by applying 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 long and that the length of side a is 3.00 ft long. You want to determine the size of angle A. Side a is opposite angle A; therefore, A table of natural functions tells you that an angle with sine 0.6 measures (to the nearest minute) 36°52'. Suppose that, instead of knowing the length of a, you know the length of b (4.00 ft). Side b is the side adjacent to angle A. You know that A table of natural functions tells you that an angle with cosine 0.8 measures 36°52'. If you know the size of one of the acute angles in a right triangle and the length of the side opposite, you can determine the length of the hypotenuse from the sine of the angle. Suppose that for the triangle shown in figure 1-23, you know that angle A = 36°52' and side a = 3.00 ft. If you know the size of one of the acute angles in a right triangle and the length of the side adjacent, you can determine the length of the hypotenuse from the cosine of the angle. Suppose that for the triangle in figure 1-23, you know that angle A = 36°52' and side b = 4.00 ft. Tables show that cos 36°52' = 0.80003. There- fore, Solution by Law of Sines For any triangle (right or oblique), when you know the lengths of two sides and the size of the angle opposite one of them, or the sizes of two angles and the length of the side opposite one of them, you can solve the triangle by applying the law of sines. The law of sines (which is explained and proved in NAVPERS 10071-B, chapter 5) states that the lengths of the sides of any triangle are proportional to the sines of their opposite angles. 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. If it follows that Figure 1-24.—Oblique triangle (law of sines). 1-21

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