CHAPTER VII. ON THE RELATIONS BETWEEN THE TRIGONOMETRICAL RATIOS OF THE SAME ANGLE. 102. The following relations are evident from the definitions : 1 1 cosec 1 sin ? cot A cos o tan A sec We have sin 0 = perpendicular base hypotenuse = tan 0. base 104. We may prove similarly cot e cos a sin 0 Or thus, coto 1 tan 0 = 105. Euclid I. 47 gives us that in any right-angled triangle the square on the hypotenuse the sum of the squares on the perpendicular and on the base, or, (hypotenuse)' = (perpendicular)* + (base)”. (i) (hypotenuse), and we get 1 = sino A + cosa 0. 2 3 = (hypotenuse ) that is, = (ii) Divide each side of the same identity by (base)', and we get hypotenuse perpendicular) base base base) 2 base\ 2 = + (iii) Divide each side of the same identity by (perpendicular), and we get hypotenuse perpendicular perpendicular) perpendicular, 2 2 106. Thus the three results (i) cos 0 + sin 0 = 1 (ii) 1 + tan' 0 = sec 0 (iii) 1+ cot' 0 = coseco 0 are each a statement in Trigonometrical language of Euc. I. 47. 107. We give the above proof in a different form. + In OE take any point P, and draw PM perpendicular to OR. Then with respect to 0, MP is the perpendicular, OP is the hypotenuse, and OM is the base; MP2 OM .. sino 0 cos? 0 = OP2 OP2 • We have to prove that sino 0 + cos 0 = 1, MP2 OM that is, that = 1, OP2 OP MP + OM OP2 i.e. that OP2 OP2) i.e. that MP + OM=OP?. But this is true by Euclid I. 47. Therefore coso A + sin? 0 =1. 1 + tan’ 0 = sec , and that 1 + coto 0 = cosec. 108. The following is a LIST OF FORMULÆ with which the student must make himself familiar : 1 cosec 0 sin Ꮎ ? 109. In proving Trigonometrical identities it is often convenient to express the other Trigonometrical Ratios in terms of the sine and cosine. Example. Prove that tan A +cot A= =sec A. cosec A. sin A Since tan A cot A= sin A' 1 1 sec A= and cosec A COS A sin A' we have to prove that sin A 1 1 sin A COS A'sin A' or that sin? A +cos? A 1 cos A, sin A COS A. sin A' and this is true, because sin? A + cos2 A=1. 2 COS A 110. Sometimes it is more convenient to express all the other Trigonometrical Ratios in terms of the sine only, or in terms of the cosine only. Example. Prove that sin4 6+2 sin8. cosa 0=1 - cost 0. Hence, putting 1 – cosa 0, and (1 - cos? 6)2 for sin? 6 and sin4 0 respectively, we have to prove that (1 - cos? 0)2 +2.(1 - cosa ). cos? A=1 - cost 0, or that 1 - 2 cos0 + cos4 6+2 cos0 - 2 cos4 A=1-cost , This example may be proved directly, by reversing the steps of the above proof; thus .: (1 – 2 cosa 0 + cos4 0) + 2 cos' 0 - 2 cos4 6=1 - cost 0, .: (1 - cosa 0)2 + 2 cos? 0 (1 - cosa 0)=1 - cos4 0, NOTE. (1 – cos 6) is called the versed sine of 0; it is abbreviated thus versin 0. |