leg to the adjacent leg increases, the angle of elevation at A also increases. Figure 1. Figure 3. Mathematicians have figured out that in right triangles of any size the ratio of the opposite leg to the adjacent leg is always the same for the same angle. If the angle is 12°, the ratio is 21: 100, or, or .21, whatever the size of the triangle. If the angle is 22°, the ratio is 40: 100, or .40. If it is 35°, the ratio is .70; and if it is 45°, the ratio is 1. This last means that both legs are equal and the triangle is isosceles. There is a special name for this ratio of the opposite opposite leg leg to the adjacent leg It is called the adjacent leg, tangent of the angle A" and is written "tan A." Mathematicians have calculated accurate tables for all these values. A simple table will be found on page 101. A 90 B 60 C Example I. For instance, suppose that in the right triangle ABC, BC is 60 feet and AC is 90 feet. We want to find the size of angle A. The tangent of the angle A (tan A) is the ratio of the opposite to the adjacent leg. We look in our table for the angle corresponding to the tangent .67 and find that it is 34°. Therefore, the angle A is 34°. Example II. Suppose we know the angle A is 20° and the opposite leg BC is 18 feet. We want to find the length of the angle of elevation and the length of the adjacent leg are known, as shown in the following. Example III. The boys saw surveyors measuring the height of a tower. They set their transit 150 feet from the foot of the tower on the level. The angle of elevation was measured and found to be 42°. Below is the simple way the surveyors worked, knowing the adjacent leg to be 150 and letting x equal the opposite leg. Example IV. If tan x = 2.08 find the angle x. As the tangent of an angle depends upon the size of the angle, it is called a function of that angle. Thus tan A is a function of A. See page 52. Table of Tangents The following table gives the values of the tangents of angles from 1 degree to 89 degrees. How Trigonometry Simplifies the New Mathematics 1. When a vertical rod 6 feet high casts a shadow 10 feet long, what is the angle of elevation of the sun? 2. What is the elevation when a 16-foot pole casts a shadow 10 feet long? 3. If the legs of a right triangle are each 8 inches, how many degrees in one of the acute angles? 62 A4 B X 4. The angle of elevation of an airplane as viewed from a point on level ground 2000 feet from the point directly under the airplane was found to be 76°. How high was the airplane? 5. To find the distance across a river to an object B, a man sighted a line AC (100 feet long) at right angles to BC at C. He measured the angle CAB and found it was 62°. How far was it across the river? Check by a scale drawing. 100 ft. THE LIGHT Review: Improving Acquaintance with Equations Solve each of the following for the value of n and check your answer. 1. n + 7 = 19. 2. n +21 3. n + 5.25 4. 2n = 16. 5. 5 n = 35. 8. 15.3 = n + 4. 9. 13 = n + 9.6. 3 = 1.745. = 41.614. 4 Checking Your Habit of Accuracy There are two things in your mathematics work which you must not lose; first, the habit of selecting right processes for problems, and second, the habit of accuracy in securing the correct solution. Suggestion to Executive Committee: In some of the following examples it may be advisable for your committee, one at a time, to explain to the class the proper method of solution. Do not work the problems; merely tell how you would proceed to solve them. Then let the pupils work independently or coöperatively by committee or team plan as you and the class decide. Be as helpful as you can in assisting classmates to strengthen the habit of accuracy. |