Plane and Spherical Trigonometry

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Macmillan, 1918 - Trigonometry - 141 pages

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Page 66 - 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 43 - 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 99 - The spherical excess of any spherical polygon is equal to the excess of the sum of its angles over two right angles taken as many times as the polygon has sides, less two.
Page 66 - 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 65 - 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 41 - 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 88 - A, 4 cos 6 = cos c cos a + sin c sin a cos B, > (1) . , "cos c = cos a cos b + sin a sin b cos C. J Whence . cos a — cos b...
Page 36 - The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.
Page 36 - The logarithm of a product is equal to the sum of the logarithms of its factors.
Page 34 - ... consists of two parts, an integral part and a decimal part. The integral part is called the characteristic of the logarithm, and may be either positive or negative.

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