19. Practical Use of the Cotangent. Since by definition we have Find b to four significant figures, given the following: 11. How far from a tree 50 ft. high must a person lie in order to see the top at an angle of elevation of 60° ? 12. From the top of a tower 300 ft. high, including the instrument, a point on the ground is observed to have an angle of depression of 35°. How far is the point from the tower? M 35° 300 13. From the extremity of the shadow cast by a church spire 150 ft. high the angle of elevation of the top is 53°. What is the length of the shadow? 14. A tree known to be 50 ft. high, standing on the bank of a stream, is observed from the opposite bank to have an angle of elevation of 20°. The angle is measured 200 50 on a line 5 ft. above the foot of the tree. How wide is the stream? 20. Practical Use of the Secant. Since by definition we have For example, given b = 15 and A= 30°, find c. Exercise 10. Use of the Secant Find c to four significant figures, given the following: 9. A ladder rests against the side of a building, and makes an angle of 28° with the ground. The foot of the ladder is 20 ft. from the building. How long is the ladder? 20 10. From a point 50 ft. from a house a wire is stretched to a window so as to make an angle of 30° with the horizontal. Find the length of the wire, assuming it to be straight. 11. In measuring the distance AB a surveyor ran the line AC, making an angle of 50° with AB, and the line BC perpendicular to AC. He measured AC and found that it was 880 ft. Required the distance AB. 12. From the extremity of the shadow cast by a tree the angle of elevation of the top is 47°. The shadow is 62 ft. 6 in. long. How far is it from the top of the tree to the extremity of the shadow? 13. The span of this roof is 40 ft., and the roof timbers AB make an angle of 40° with the horizontal. Find the length of AB. B 140° 40 ft. 21. Practical Use of the Cosecant. Since by definition we have Exercise 11. Use of the Cosecant Find e to four significant figures, given the following: B la 3. α = = 36, 11. Seen from a point on the ground the angle of elevation of an aeroplane is 64°. If the aeroplane is 1000 ft. above the ground, how far is it in a straight line from the observer ? 12. A ship sailing 47° east of north changes its latitude 28 mi. in 3 hr. What is its rate of sailing per hour? 13. A ship sailing 63° east of south changes its latitude 45 mi. in 5 hr. What is its rate of sailing per hour? 14. From the top of a lighthouse 100 ft., including the instrument, above the level of the sea a boat is observed under an angle of depression of 22°. How far is the boat from the point of observation? 15. Seen from a point on the ground the angle of elevation of the top of a telegraph pole 27 ft. high is 28°. How far is it from the point of observation to the top of the pole? 16. What is the length of the hypotenuse of a right triangle of which one side is 11 in. and the opposite angle 43°? 22. Functions as Lines. The functions of an angle, being ratios, are numbers; but we may represent them by lines if we first choose a unit of length, and then construct right tri angles, such that the denominators of the ratios shall be equal to this unit. Thus in the annexed figure the radius is taken as 1, the circle then being spoken of as a unit circle. Then N B A M Learn In each case we have arranged the fraction so that the denominator is 1. This explains the use of the names tangent and secant, AT being a tangent to the circle, and OT being a secant. Formerly the functions were considered as lines instead of ratios and received their names at that time. The word sine is from the Latin sinus, a translation of an Arabic term for this function. We see from the figure that the sine of the complement of a is NP, which equals OM; also that the tangent of the complement of x is BS, and that the secant of the complement of x is OS. Exercise 12. Functions as Lines 1. Represent by lines the functions of 45°. 2. Represent by lines the functions of an acute angle greater than 45°. Using the above figure, determine which is the greater : 3. sin x or tan x. 5. secx or tan x. 4. sin x or sec x. 6. cscx or cot x. 7. cos x or cot x. 15. Show that the sine of an angle is equal to one half the chord of twice the angle in a unit circle. 16. Find x if sin x is equal to one half the side of a regular decagon inscribed in a unit circle. Given x and y, x + y being less than 90°, construct a line equal to 32. Show by construction that 2 sin A >sin 2 A, when A<45°. 33. Show by construction that cos A< 2 cos 2 A, when A< 30°. 34. Given two angles A and B, A+B being less than 90°; show that sin(A+B) < sin A+ sin B. 35. Given sin x in a unit circle; find the length of a line in a circle of radius r corresponding in position to sin x. 36. In a right triangle, given the hypotenuse c, and sin A=m; find the two sides. 37. In a right triangle, given the side b, and tan A=m; find the other side and the hypotenuse. Construct, or show that it is impossible to construct, the angle x, given the following: 47. Using a protractor, draw the figure to show that sin 60° = cos (of 60°), and sin 30° = cos (2 × 30°). |