Lesson 9 Tangents The tangent of an angle is defined as the ratio of the side opposite the angle to the side adjacent the angle. Thus, if you will refer to figure B on page 52, the tangent of angle A is written as tan After you have finished this lesson, you will be able to find all of the parts of a right triangle if you are given either one side and an acute angle or two sides. Text Assignment Chapter X, pages 177-192. Omit "Basis of the Process," pages 179-180. Study Notes Page 179. There are several Practice Problems on this page where you should obtain more accurate answers than those given in the text. Your answers to Practice Problems 3, 7, 9, and 12 should be 0.314, 0.369, 0.767, and 5.51 inches, respectively. Pages 180-182. The second paragraph on this page explains how to obtain the tangent of an angle between 1° and 5°43'. As your text explains on page 182, the tangent of a small angle is very nearly equal to the angle in radians. This very same statement was made in the last lesson about the sine of a small angle. You can see this is a valid statement by examining figure E on page 55 again and imagining angle A to be very small. If angle A is very small (less than 5°43′), then the arc length and opposite side are almost equal in length. Similarly, the adjacent side and radius are almost equal in length. This enables us to develop another method for finding the opp. tangent of an angle between 0°34′ and 5°43'. Since sin A= hyp. tan A= 2; and since the hypotenuse (radius) and adjacent side opp.. are almost equal for small angles, then for these angles sin A= tan A. In view of the foregoing, you can find the tangent of an angle between 0°34′ and 5°43' by simply using the sine scale on the back of your slide rule. For angles less than 0°34', also determine them by the same method as for finding the sine of such angles. Change the angle from minutes and seconds to minutes and a decimal fraction of a minute. When the angle is expressed in this manner, divide by 3,440. Page 182. Your answer to Practice Problem 6 will probably be 0.002385. In Practice Problem 7 where you are given 12′25.5′′, you must divide 25.5" by 60 to get 0.425'. Then to find the tangent of 12.425' divide by 3,440. The answer to Practice Problem 9 is 0.00505. Also, your answer to Practice Problem 10 will probably be 0.0173. 1 For angles between 45° and 90° the tangent can be found by one simple procedure. Find the tangent of 90° minus the angle and then use the relationship tan @= You can easily prove this relationship to yourself if you look at figure B on page 52 again. The tangent of angle A is expressed as the ratio of one length to another. The tangent of angle B, which is (90°-A), is expressed as the ratio of the same lengths, only inverted. Thus in figure B, tan A=; 1 tan B To summarize, there are five different procedures for finding the tangent of an angle: 1. For finding the tangent of an angle between 0° and 0°34' 2. For finding the tangent of an angle between 0°34′ and 5°43′ 3. For finding the tangent of an angle between 5°43′ and 45° 4. For finding the tangent of an angle between 45° and 84°17' 5. For finding the tangent of an angle between 84°17' and 90° Care must be taken to differentiate between procedures 3 and 4. The tangent of angles between 5°43′ and 45° are read on the C scale above the right index of the D scale. Tangents of angles between 45° and 84°17′ are read on the D scale under the left index of the C scale. Your answer to Illustrative Example 7 will probably be closer to 2.14 than 2.15 as given in the text. Page 183. The answer to Practice Problem 7 is 2.00 instead of 2.01. Pages 184-186. Remember, the tangent of an angle between 0° and 90° and between 180° and 270° has a positive value. The tangent of an angle between 90° and 180° and between 270° and 360° has a negative value. Page 186. The answers to Practice Problems 4, 6, and 9 are -0.246, -0.700, and 0.521, respectively. Pages 186-187. The answers to Illustrative Examples 17 and 18 are not very precise. These answers are 32°52′ and 11°28′, respectively. Page 187. Slightly more accurate answers for Practice Problems 4, 5, and 8 are 18°25′, 35°15′, and 25°2′, respectively. Since the arc tangent of 1 is 45° (the opposite side is equal to the adjacent side), the arc tangent of any number greater than 1 is going to be an angle between 45° and 90°. Page 188. The answers to Practice Problems 8 and 10 should be changed to 64°51′ and 53°28′, respectively. You will probably get 2°9' instead of 2°10' as your answer to Illustrative Example 23 since 0.0376X57.3 is 2.15 rather than 2.16. Page 190. There are two Practice Problems in which your answers should be slightly more precise than those given in the text. The answers to Practice Problems 14 and 25 are 0.946 and 56°45', respectively. Page 192. The answers to these Review Problems are given at the back of this study guide. No answers are required for problems 26-30. Again we suggest you make sketches for problems such as 1, 2, and 5. In problem 3, find the angle first and then find the tangent of the angle. Self-Examination Select the correct answer in each of the following exercises. 13. A man standing on level ground 475 feet from a vertical wall looks at an object 367 feet up the wall. What angle does his line of sight make with the ground? a. 7°21' b. 37°40' c. 52°20' d. 82°39' e. None of the above 14. A man riding in a car approaches a right angle intersection. When he is still 597 feet from the intersection he looks 15° to his left and spots another car approaching on the crossroad. How many feet from the intersection is the second car? a. 160 b. 223 c. 430 d. 823 e. None of the above 15. A man runs a piece of twine from the top of a 148 foot building until it touches the ground at an angle of 76°30'. How many feet is it from the point where the twine touches the ground to the base of the building? 1. Using examples in your text as a guide, explain, step by step, how you would determine the tangent of 39°15' using the slide rule. 2. Using examples in your text as a guide, explain, step by step, how you would determine the tangent of 59°39' using the slide rule. 3. Using examples in your text as a guide, explain, step by step, how you would determine the angle which has a tangent equal to 0.00329. 629329-62 (Continued on next page) |