2. Show that in the right triangle ACB (§ 7) the following Find the values of the six functions of A, if a, b, c respectively have the following values: 7. 3, 4, 5, 8. 5, 12, 13. 9. 8, 15, 17. 10. 9, 40, 41. 11. 3.9, 8, 8.9. 13. What condition must be fulfilled by the lengths of the three lines a, b, c (§ 7) to make them the sides of a right triangle? Show that this condition is fulfilled in Exs. 7-12. Find the values of the six functions of A, if a, b, c respectively have the following values: 18. As in Ex. 13, show that the condition for a right triangle is fulfilled in Exs. 14-17. Given a2 + b2 = c', find the six functions of A when: 19. a = b. 20. a 2b. 21. a = fc. Given a2 + b2 = c2, find the six functions of B when : Given a2 + b2 = c2, find the six functions of A and also the six functions of B when: 2 26. a = √p2 + q2, b = √2 pq. 27. a = √p2+p, c = p +1. of 6 PLANE TRIGONOMETRY In the right triangle ACB, as shown in § 7: = , and c = 20.5. 0.44, and c = 3.5. 2음. 5 ༢...--. = 20.5 28. Find the length of side a if sin A = 33, and b Find the hypotenuse and other side of a right triangle, given : 0.6, and 38. The hypotenuse of a right triangle is 2.5 mi., sin A = cos A 0.8. Compute the sides of the triangle. = 39. Construct with a protractor the angles 20°, 40°; and 70°; determine their functions by measuring the necessary lines and compare the values obtained in this way with the more nearly correct values given in the following table: Find, by means of the above table, the sides and hypotenuse of a 55. By dividing the length of a vertical rod by the length of its horizontal shadow, the tangent of the angle of elevation of the sun at that time was found to be 0.82. How high is a tower, if the length of its horizontal shadow at the same time is 174.3 yd.? 56. A pin is stuck upright on a table top and extends upward 1 in. above the surface. When its shadow is in. long, what is the tangent of the angle of elevation of the sun? How high is a tele graph pole whose horizontal shadow at that instant is 21 ft. ? 8. Functions of Complementary Angles. In the annexed figure we see that B is the complement of A; that is, B = 90° - A. Hence, That is, each function of an acute angle is equal to the co-named function of the complementary angle. Co-sine means complement's sine, and similarly for the other co-functions. It is therefore seen that sin 75°: = cos (90° 75°) = cos 15°, sec 82° 30' csc (90° 82° 30')= csc 7° 30', and so on. Therefore, any function of an angle between 45° and 90° may be found by taking the co-named function of the complementary angle, which is between 0° and 45°. Hence, we need never have a direct table of functions beyond 45°. We shall presently see (§ 12) that this is of great advantage. Exercise 2. Functions of Complementary Angles Express as functions of the complementary angle: Express as functions of an angle less than 45° : 16. cos 88° 10'. 12. sec 45°. PLANE TRIGONOMETRY 9. Functions of 45°. The functions of certain angles, among them 45°, are easily found. In the isosceles right triangle ACB we have AB 45°, and a b. Furthermore, since a+b2c2, we have 2 a2 = c2, a √2 = c, and a = c √2. Hence, = B a tan 45° =cot 45° We have therefore found all six functions of 45°. For purposes of computation these are commonly expressed as decimal fractions. Since √2 = 1.4142 +, we have the following values: 10. Functions of 30° and 60°. In the equilateral triangle AA'B here shown, BC is the perpendicular bisector of the base. Also, b = §c, and a = √ c2 − b2 = √ c2 − ↓ c2 = †c √3. Hence, - α = 810 2018 b α = α = = 1 √3 = 1 The sine and cosine of 30°, 45°, and 60° are easily remembered, thus: The functions of other angles are not so easily computed. The computation requires a study of series and is explained in more advanced works on mathematics. For the present we assume that the functions of all angles have been computed and are available, as is really the case. Exercise 3. Functions of 30°, 45°, and 60° Given √3 = 1.7320, express as decimal fractions the following Write the ratios of the following, simplifying the results : 43. By what must sin 45° be multiplied to equal tan 30° ? |