Artículos relacionados a Elementary trigonometry, plane and spherical
Sinopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1889 Excerpt: ... the angle between O X and the positive direction of Q R, which is R Q; and that for the side O R is A--B. The projections of O Q are oQ' =oq cos A Oq" = Oq sin A or, by substituting for OQ its equivalent, OR cos B, as above given. o Q' = o R cos A cos B o Q" = o R sin A cos B The projections of QR are Q' R' =qr cos (4 + 90)=--Qr sin A Q" R" = Q R sin (A + 90) = Q R cos A or, by substituting for QR its equivalent,-O R sin B, as above given. Q' R' = OR sin A sin B Q"r" =--Orcos A sin B The projections of O R are Or' =or cos (A--B) Or" = Or sin (A--B) The projection of one side of a triangle being equal to the sum of the projections of the other two sides, we have 58 the two equations Or" = Oq"+q"r" O R' = O Q' + Q' R' which, by substituting the equivalents above found and dividing through by O R, become sin (A--B) = sin A cos B--cos A sin B... 63 cos (A--B) = cos A cos B + sin A sin B.., 64 That this proof is perfectly general may be shown by drawing the figure in different forms and repeating the proof with each form, as was suggested in Article 147. 150. The striking similarity between the proof just given and that given for the formulas 61 and 62 arises from the fact that the two proofs are, in so far as the reasoning is concerned, identical, the only differences being certain differences of expression occasioned by the use of the negative angle--B instead of the positive angle + B. Indeed, formulas 63 and 64 are simply what 61 and 62 become by the substitution of--B for + B therein. 151. This fact suggests that the geometrical proof of 63 and 64 may be set aside as unnecessary, and that these formulas may be derived from 61 and 62 by a purely analytic process. The following is such a process. Assumin...
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Reseña del editor
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1889 Excerpt: ... the angle between O X and the positive direction of Q R, which is R Q; and that for the side O R is A--B. The projections of O Q are oQ' =oq cos A Oq" = Oq sin A or, by substituting for OQ its equivalent, OR cos B, as above given. o Q' = o R cos A cos B o Q" = o R sin A cos B The projections of QR are Q' R' =qr cos (4 + 90)=--Qr sin A Q" R" = Q R sin (A + 90) = Q R cos A or, by substituting for QR its equivalent,-O R sin B, as above given. Q' R' = OR sin A sin B Q"r" =--Orcos A sin B The projections of O R are Or' =or cos (A--B) Or" = Or sin (A--B) The projection of one side of a triangle being equal to the sum of the projections of the other two sides, we have 58 the two equations Or" = Oq"+q"r" O R' = O Q' + Q' R' which, by substituting the equivalents above found and dividing through by O R, become sin (A--B) = sin A cos B--cos A sin B... 63 cos (A--B) = cos A cos B + sin A sin B.., 64 That this proof is perfectly general may be shown by drawing the figure in different forms and repeating the proof with each form, as was suggested in Article 147. 150. The striking similarity between the proof just given and that given for the formulas 61 and 62 arises from the fact that the two proofs are, in so far as the reasoning is concerned, identical, the only differences being certain differences of expression occasioned by the use of the negative angle--B instead of the positive angle + B. Indeed, formulas 63 and 64 are simply what 61 and 62 become by the substitution of--B for + B therein. 151. This fact suggests that the geometrical proof of 63 and 64 may be set aside as unnecessary, and that these formulas may be derived from 61 and 62 by a purely analytic process. The following is such a process. Assumin...
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