7. Find the edge length in inches of cubes having the following volumes: a. 118 cu. in. b. 2.84 cu. ft. c. 0.067 cu. yd. d. 0.83 cu. ft. e. 54,500 cu. in. 8. The center deflection or sag of a wooden beam 4X6 inches in cross section is given by the formula D= Pr 5,520,000,000' where D is the deflection in inches, P is the load in pounds placed at the center of the beam, and is the length of the beam between supports. This length must be in inches. What center deflection would such a beam have with the following lengths and loads? a. A 14 ft. beam with a 300 lb. center load Lesson 8 Sines and Cosines While it is not necessary to know what is meant by the sine or cosine of an angle in order to obtain their correct value from the slide rule, a simple explanation of these functions of an angle will be helpful if you are ever called upon to solve a problem where it is not definitely stated which of these functions you are to use. The drawing below is of a right triangle. As pointed out in an earlier lesson, this is a right triangle which contains a 90° angle or right angle. The angles are A, B, and C, with C being the right angle. The sides are a, b, and c. Note that side a is opposite angle A, side b opposite angle B, and side c opposite angle C. Now let us consider angle A where the arrows have been drawn. The side across from The shorter side next to angle A is called the opposite side of A. angle A is called the adjacent side of A. were to consider angle B, we would find the opposite side of B to be side b and the adjacent side to be side a. Regardless of which angle we are considering, side C is always the hypotenuse. The sine of an angle is the ratio of the side opposite the angle to the hypotenuse. Thus, the sine (abbreviated sin) of angle A is written opp. α as follows: sin A= As angle A becomes smaller and side c remains constant, side a also becomes smaller; therefore, the sine of angle A becomes smaller. When angle A becomes zero, the sine of angle A becomes zero. On the other hand, as angle A increases and side c again remains constant, side a also becomes larger; therefore, the sine of angle A becomes larger. When angle A becomes 90°, side a coincides with side c and the sine of angle A becomes equal to 1. Thus, the sine of any angle always has an absolute value between zero and 1. You can illustrate this by sketching several angles of different sizes. adj. b hyp. c The cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse. Thus, the cosine (abbreviated cos) of angle A is written as follows: cos A=; As angle A becomes smaller and side c remains constant, side b becomes larger; therefore, the cosine of angle A becomes larger. When angle A becomes zero, sides b and c coincide and the cosine of angle A becomes equal to 1. As angle A increases and side c remains constant, side b becomes smaller; therefore, the cosine of angle A becomes smaller. When angle A becomes 90°, the cosine of angle A becomes zero. To summarize, as angle A increases from 0° to 90°, the sine of angle A increases from 0 to 1 and the cosine of angle A decreases from 1 to 0. The values of the sines and cosines of an angle depend only on the size of the angle under consideration. Variation in the lengths of the sides in the triangle will not change the value of the sine or cosine of an angle if the size of the angle is unchanged. Thus in figure C, the sine of angle A in the triangle on the left is equal to the sine of of angle A' in the triangle on the right. There are four other functions of angles besides the sine and cosine. The names of these, with their abbreviations in parentheses, are tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). The tangent function will be discussed in the next lesson. These six func tions are called the trigonometric functions and are the basis for trigonometry. Courses in trigonometry at both the high school and college levels are offered by USAFI if you are interested in pursuing this subject further. Text Assignment Chapter IX, pages 157-160, 161-172, 173-175. Omit "Basis of the Process," pages 160-161, and "Basis of the Process," page 172. Study Notes Before studying this chapter, it would be wise to review pages 28-36 in your text. These pages will refresh your memory on the reading of angles on the sine scale. You may find it a little difficult to get used to reading the sine scale where the degrees are divided into minutes after studying several lessons in which you used the A, B, C, and D scales. Page 157. If you are using the slide rule furnished by USAFI, you will find a celluloid insert on the reverse side. This celluloid insert contains a hairline which is used to locate numbers on the sine, log, and tangent scales. We shall confine this lesson to the sine scale which is identified on the slide rule by the letter S. All directions in the study guide will assume your slide rule contains a celluloid insert with a hairline. The angle in degrees is set on the S scale under the hairline. The sine of this angle is read on the front side of the slide rule on the B scale under the right index of the A scale. Remember, the absolute value of the sine of an angle cannot be less than 0 or greater than 1. The left index of the B scale has the value 0.01, the center index has the value 0.1, and the right index the value 1. To become accustomed to reading these values, be certain to work through the Illustrative Examples on pages 158-160 and the Practice Problems on page 160. Page 160. In working Practice Problems 11 and 12 it will be helpful to make sketches such as shown in figure D. When solving these problems you must remember that the sine of an angle is equal to the ratio of the side opposite the angle to the hypotenuse. In Practice Problem 11: sin 40°- or x=230 sin 40°. Solving for x we get 230 Similarly, in Practice Problem 12: sin 22°30′ Solving for x we get x=47 pounds. You will need pencil and paper to write down the numbers in Practice Problems 13 and 14 since you will be unable to work these problems without transferring numbers from one scale to another. In Practice Problems 16 and 17, sin2 may be new to you. The expression sin2 40° is read the square of the sine of 40°. In other words, you determine the sine of 40° and square this value. Pages 160-161. Since an angle of 0°34′ is very small, the side opposite this angle in a triangle will also be very small. In fact, it is only 0.01 times the length of the hypotenuse, as your slide rule indicates. Because the sine scale on the slide rule does not go below 0°34', your text describes a special method for calculating the sines of smaller angles. Angles can be measured in a unit called radians as well as in degrees. This is a very simple unit of measure in that an angle of 1 radian has an arc length equal to the radius of the arc. Thus, to find the size of an angle in radians, measure the length of the arc and divide it by the radius. With a ruler and a protractor you can show that an angle of 57.3° has an arc length equal to its radius and, therefore, equal to 1 radian. As angle A in figure E below becomes very small the arc length and opposite side (broken line) become very nearly the same length. When this occurs, the sine of angle A (opposite side divided by hypotenuse) is almost the same as the angle in radians (arc length divided by radius). For this reason, when you wish to find the sine |