Plane and Spherical Trigonometry

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Macmillan Company, 1918 - Trigonometry - 141 pages
 

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Page 68 - In any triangle the sides are proportional to the sines of the opposite angles. That is, Fig. 25, a : b : c = sin A : sin B : sin C.
Page 71 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
Page 98 - A cos 6 = cos a cos c + sin a sin c cos B cos c = cos a cos 6 + sin a sin 6 cos C Law of Cosines for Angles cos A = — cos B...
Page 73 - In any triangle, the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides, minus twice the product of one of these sides and the projection of the other side upon it.
Page 97 - B . sin c = sin b . sin C cos a = cos b . cos c + sin b . sin c cos b = cos a . cos c + sin a . sin c cos A cos B cos c = cos a . cos b + sin a . sin b . cos C ..2), cotg b . sin c = cos G.
Page 48 - ... cos y + cos x sin y cos x cos y — sin x sin y tan a- + tan y 1 — tan x tan y sin (x — y) = sin x cos y — cos x...
Page 98 - The law of sines states that in any spherical triangle the sines of the sides are proportional to the sines of their opposite angles: sin a _ sin b __ sin c _ sin A sin B sin C...
Page 99 - VI — cos2 a — cos2 6 — cos2 c+2 cos a cos b cos c sin a sin a sin b sin c where the positive sign is taken because A and a are each less than 180░.
Page 96 - AB'C, b < 90░ and c' < 90░, and, therefore, cos a' = cos b cos c' + sin b sin c' cos CAB'. But a' = 180░ - a, c'= 180░ - c, and CAB' = 180░ - A. Hence cos (180░ - a) or, cos a = cos b cos c + sin 6 sin c cos A, which proves the law of cosines for all cases.
Page 107 - ... enumerated in the solution of oblique spherical triangles. 1. Given the three sides, a, b, c. 2. Given the three angles, A, B, C. 3. Given two sides and the included angle, a, b, C. 4. Given two angles and the included side, A, B, c. 5. Given two sides and the angle opposite one of them, a, b,A.

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