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Tables of decimal values for the trigonometric ratios may be constructed in a variety of ways. Some give the angles in degrees, minutes, and seconds; others in degrees and tenths of a degree. The latter method is more compact and is the method used for appendix II. The "headings" at the bottom of each page in appendix II provide a convenient reference showing the minute equivalents for the decimal fractions of a degree. For example, 12 (12 minutes) is the equivalent of 0.2.

Finding the Function Value

The trigonometric ratios are sometimes called FUNCTIONS, because the value of the ratio depends upon (is a function of) the angle size. Finding the function value in appendix II is easily accomplished. For example, the sine 35 is found by looking in the "sin" row opposite the large number 35, which is located in the extreme left-hand column.

Since our angle in this example is exactly 35, we look for the decimal value of -the sine in the column with the 0.0 heading. This column contains decimal values for functions of the angle plus 0.0; in our example, 35 plus 0.0, or simply 35.0. Thus we find that the sine of 35.0 is 0.5736. By the same reasoning, the sine of 42.7 is 0.6782, and the tangent of 32.3 is 0.6322.

A typical problem in trigonometry is to find the value of an unknown side in a right triangle when only one side and one acute angle are known. EXAMPLE: In triangle ABC (fig. 19-8), find the length of AC if AB is 13 units long and angle CAB is 34.7.

Figure 19-8.-Using the trigonometric ratios to evaluate the sides.


The angles of a triangle are frequently stated in degrees and minutes, rather than degrees and tenths. For example, in the foregoing problem, the angle might have been stated as 34 42. When the stated number of minutes is an exact multiple of 6 minutes, the minute entries at the bottom of each page in appendix II may be used.

Finding the Angle

Problems are frequently encountered in which two sides are known, in a right triangle, but neither of the acute angles is known. For example, by applying the Pythagorean Theorem we can verify that the triangle in figure 19-9 is

Figure 19-9.-Using trigonometric ratios to evaluate angles.

a right triangle, The only information given, concerning angle q, is the ratio of sides in the triangle. The size of q is calculated as follows:

Assuming that the sides and angles in figure 19-9 are in approximately the correct proportions, we estimate that angle q is about 20. The table entries for the tangent in the vicinity of 20 are slightly too small, since we need a number near 0.4167. However, the tangent of 22 36 is 0.4163 and the tangent of 22 42 is 0.4183. Therefore, q is. between 22 36 and 22 42 . 

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