Logarithm table.-(1) 82. Application to right triangles.-a. Table IV of appendix II gives common logarithms of circular or trigometric functions and will be used to solve trigonometric problems by logarithms. It is to be understood that a (-10) must be supplied whenever a characteristic of 8 or 9 is given in a log column. Interpolation is applied as in common logarithms. (2) Corresponding to 18°, one finds log sin 18°=9.4980-10. For 62°, it is found that log tan 62°=.2743. Again, by interpolation, log cos 7.7° 9.9961-10. b. Solving right triangles.-The following process gives the unknown parts of a right triangle: By use of the definition of the sine, cosine, tangent, and cotangent functions, write an equation which involves just one missing part; then solve this equation for the missing part, and perform the indicated arithmetic by logarithms. Example 1: Given A=62°10′, a=78; find B, b, and c. Example 2: Given a=40, c=59.3; find A, B, and b. (Since the pythagorean relation' is not well adapted to logarithms, find the angles first, and then find b by a trigonometric relation.) 83. Evaluating formulas with logarithms.-a. There has been given a number of formulas for calculating areas and volumes of various objects. By solving the equation of the formula for a missing part, logarithms can often be used to compute the missing part when the others are given. Example 1: Find the volume of a sphere whose radius is 6 feet. Use T=3.1416, or log π=.4971 1 The relation of the hypotenuse to the legs of a right triangle, namely that the square of the hypotenuse is equal to the sum of the squares of the legs. Example 2: A right circular cone has an altitude of 18 inches and a volume of 364 cubic inches. What is the radius of its base? b. Exercises. Using the conventional triangle, solve for the missing parts of the following right triangles, finding sides to four significant digits and angles to tenths of degrees or minutes. (1) A=23.5°, c=627 (2) B=76°15', c=93.4 A=13°45′, a=22.20, b=90.7 Answer. A=22°, b=180.7, c=194.9 Answer. (6) a=21.9, c=91.9 A=13°47'1" B=76°13′ b=89.25 Answer. (7) b=18.3, c=30.75 (8) A tin can has base diameter 4.5 inches and height 5 inches. What is its cubic capacity? 79.52 cubic in. Answer. (9) What is the radius of a sphere whose volume is 700 cubic feet? (10) What is the volume of a sphere whose radius is twice that of the sphere in the preceding problem? 5,600 Answer. (11) Find the angle at the vertex of the cone discussed in example 2, a above. 84. Oblique triangles.-a. It is often possible to find the missing parts of oblique triangles by constructing auxiliary lines, generally altitudes, which will allow the use of the theory developed for right triangles. Example 1: Find side b of the triangle shown in figure 127. Construct the altitude, BD, to the side AC, and two right triangles are formed: ABD and BDC. (1) In triangle ABD, cos A=AD, and AD=c cos A=20.95 cos (3) b=AD÷DC=19.62+13.85, b=33.47 Example 2: Find sides a and b of the triangle shown in figure 128. |