566 CHAPTER 5 Trigonometric Functions Significant Digits A number that represents the result of counting, or that results from theoretical work and is not the result of measurement, is an exact number. There are 50 states in the U.S. In this statement, 50 is an exact number. Most values obtained for trigonometric applications are measured values that are not exact. Suppose we quickly measure a room as 15 ft by 18 ft. See Figure 44. We calculate the length of a diagonal of the room as follows. d2 = 152 + 182 Pythagorean theorem d2 = 549 Apply the exponents and add. d = 2549 Square root property; Choose the positive root. d ≈23.430749 Should this answer be given as the length of the diagonal of the room? Of course not. The number 23.430749 contains six decimal places, while the original data of 15 ft and 18 ft are accurate only to the nearest foot. The results of a calculation can be no more accurate than the least accurate number in the calculation. Thus, the diagonal of the 15-by-18-ft room is approximately 23 ft. If a wall measured to the nearest foot is 18 ft long, this actually means that the wall has length between 17.5 ft and 18.5 ft. If the wall is measured more accurately as 18.3 ft long, then its length is really between 18.25 ft and 18.35 ft. The results of physical measurement are only approximately accurate and depend on the precision of the measuring instrument as well as the aptness of the observer. The digits obtained by actual measurement are significant digits. The measurement 18 ft is said to have two significant digits; 18.3 ft has three significant digits. In the following numbers, the significant digits are identified in color. 408 21.5 18.00 6.700 0.0025 0.09810 7300 Notice the following. • 18.00 has four significant digits. The zeros in this number represent measured digits accurate to the nearest hundredth. • The number 0.0025 has only two significant digits, 2 and 5, because the zeros here are used only to locate the decimal point. • The number 7300 causes some confusion because it is impossible to determine whether the zeros are measured values. The number 7300 may have two, three, or four significant digits. When presented with this situation, we assume that the zeros are not significant, unless the context of the problem indicates otherwise. To determine the number of significant digits for answers in applications of angle measure, use the following table. Significant Digits for Angles Angle Measure to Nearest Examples Write Answer to This Number of Significant Digits Degree 62°, 36° two Ten minutes, or nearest tenth of a degree 52° 30′, 60.4° three Minute, or nearest hundredth of a degree 81° 48′, 71.25° four Ten seconds, or nearest thousandth of a degree 10° 52′ 20″, 21.264° five 15 ft 18 ft d Figure 44
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