567 5.4 Solutions and Applications of Right Triangles To perform calculations with measured numbers, start by identifying the number with the least number of significant digits. Round the final answer to the same number of significant digits as this number. Remember that the answer is no more accurate than the least accurate number in the calculation. Solving Triangles To solve a triangle means to find the measures of all the angles and sides of the triangle. As shown in Figure 45, we use a to represent the length of the side opposite angle A, b for the length of the side opposite angle B, and so on. In a right triangle, the letter c is reserved for the hypotenuse. A C B a b c When we are solving triangles, a labeled sketch is an important aid. Figure 45 EXAMPLE 1 Solving a RightTriangle Given an Angle and a Side Solve right triangle ABC, if A = 34° 30′ and c = 12.7 in. SOLUTION To solve the triangle, find the measures of the remaining sides and angles. See Figure 46. To find the value of a, use a trigonometric function involving the known values of angle A and side c. Because the sine of angle A is given by the quotient of the side opposite A and the hypotenuse, use sin A. sin A = a c sin A = side opposite hypotenuse sin 34° 30′ = a 12.7 A = 34° 30′, c = 12.7 a = 12.7 sin 34° 30′ Multiply by 12.7 and rewrite. a = 12.7 sin 34.5° Convert to decimal degrees. a ≈12.710.566406242 Use a calculator. a ≈7.19 in. Three significant digits To find the value of b, we could substitute the value of a just calculated and the given value of c in the Pythagorean theorem. It is better, however, to use the information given in the problem rather than a result just calculated. If an error is made in finding a, then b also would be incorrect. And, rounding more than once may cause the result to be less accurate. To find b, use cos A. cos A = b c cos A = side adjacent hypotenuse cos 34° 30′ = b 12.7 A = 34° 30′, c = 12.7 b = 12.7 cos 34° 30′ Multiply by 12.7 and rewrite. b ≈10.5 in. Three significant digits Once b is found, the Pythagorean theorem can be used to verify the results. 348 309 c = 12.7 in. A C B a b Figure 46
RkJQdWJsaXNoZXIy NjM5ODQ=