PLANE TRIGONOMETRY. MISCELLANEOUS EXAMPLES. 1. Given A 45° 4', B 75° 35', c 457, to find the other parts. C 59° 21'. Ans. a 376.06. b 514.48. 2. Given a 454, c 753, A 45° 25', to find the other parts. 3. Given a 57, b 89, C 75° 4', to find the other parts. c 92.495. 4. Given a 41, b 74, c 63, to find the other parts. 5. Given h 75, a 35, to find the other parts. C 58° 18′ 22′′. b 66.332. 6. Given h 919, A 37° 37', to find the other parts. Ans. 7. Given b 45.3, A 34° 23′, to find a and h. 8. Given a 40, b 57, c 97, to find the other parts. h 54.890. 9. Given a 0.05377, b 0.06607, A 45°, to find the other 'parts. B 60° 19' 34", B 119° 40′ 26′′. Ans. C 74° 40′ 26′′, c 0.07334, 10. Given a 54, b 35, B 97° 15′, to find the other parts. CHAPTER IV. TRIGONOMETRIC FUNCTIONS. ANALYTICAL METHOD.* DEFINITIONS. 57. Instead of considering the Sine, Tangent, &c. as lines,. having a certain position in a circle, and varying not only as the arc, but also as the radius, we consider them, in this system, as ratios, varying only as the angle, and capable of being represented by certain lines in a circle only when the radius is unity. 58. The Sine of an angle is the ratio of the side opposite it in a right-angled triangle to the hypothenuse. That is, if in any right-angled triangle ABC we represent the hypothenuse by h, and the sides opposite the angles A and B by A a and b respectively, B h α b (1) 59. The Tangent of an angle is the ratio of the side opposite it in a right-angled triangle to the side adjacent. 60. The Secant of an angle is the ratio of the hypothenuse to the side adjacent to the angle. 61. The Cosine, Cotangent, Cosecant of an angle are respectively the sine, tangent, and secant of its complement. * Those who have taken the Geometrical Method can omit Chapters IV. and V. PLANE TRIGONOMETRY. Therefore, as the acute angles of a right-angled triangle are complements of each other, we shall have 810 218 2180 b cosec. A sec. B cosec. B sec. h 62. By inspecting these equations it will be seen that the sine and cosecant of an angle are reciprocals of each other; so also the cosine and secant, and the tangent and cotangent. That is 63. The sine, cosine, &c., vary only as the angle; that is, for a given angle they are constant. Let A DE and A B C be any two right-angled triangles, having a common angle A; they are equiangular · and similar. B D Hence A E DE: DABC: BA, DE BC or sin. A BA that is, the sine of the angle A is constant, whatever the length of the sides. In the same way it can be proved that the cosine, tangent, &c. of a given angle are constant. angle a 64. The sine, cosine, &c. can be represented by certain lines in a circle, when the radius is unity. Let CDF be a triangle, right-angled at F. With C as a centre and CD as radius, describe a circle BAE; produce CF to B, and draw BI parallel to F D, and meeting CD produced; draw CA perpendicular, and DH and AK parallel to CB. In the right-angled triangle CDF E I A K H D B F If CA, CD, CB, that is, radius, becomes unity, we shall In the Geometrical Method, without limiting the radius to unity, these lines are defined as the sine, cosine, &c. of the arc or angle to which they belong. 65. In the right-angled triangle A B C (Art. 58) PLANE TRIGONOMETRY. (9) If, therefore, the sine of an angle is known, the cosine can be found by (8), the tangent by (9), then the cotangent, the secant, and cosecant by (5). 66. Problem. To find the sine and cosine of the sum and difference of two angles, when their sines and cosines are known. HCL A — B = From the points D and H, equally 04 distant from C, draw D I and H L per IK L pendicular to CL; join D H and draw F K perpendicular, and FE and H G parallel to C L. The triangles CDF and C F H are equal. For by construction CD and CH and the angles D C F and FCH are equal, and CF is common; therefore D F is equal * In the Analytical Method these numbers standing alone in parentheses refer to the equations with the same number. |
