Plane and Spherical Trigonometry

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D.C. Heath & Company, 1917 - Trigonometry - 313 pages

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Page 129 - 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 111 - From the top of a hill the angles of depression of two successive milestones, on a straight level road leading to the hill, are observed to be 5░ and 15░.
Page 130 - A'BC, b' and c' are less than 90░. FIG. 34. The law of cosines is, therefore, true for the triangle A'BC, so that, since A' = A, cos a = cos b' cos c' + sin 6' sin c
Page 85 - B is negative, and BD — — a cos B. The substitution of this in (4) leads us again to (3). Thus we see that (3) is true in all cases. THE LAW OF COSINES. The square of any side .of a plane triangle is equal to the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. This may be regarded as a generalization of the Pythagorean Theorem to which it reduces when the included angle is a right angle. These two laws are among the most important of...
Page 131 - 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 145 - PM = sm b' = sm c, - = coa c> - = cos * 5 substituting these values, we have, cos a — sin b sin c cos A = cos J cos c ; and by transposing, cos a = cos b cos c + sin 6 sin c cos A.
Page 70 - The sum of the sines of two angles is equal to twice the product of the sine of half the sum, and the cosine of half the difference of the angles.
Page 111 - From the top of a cliff 150 ft. high the angles of depression of the top and bottom of a tower are 30░ and 60░, respectively.
Page 111 - From the top C of a cliff 600 feet high, the angle of elevation of a balloon B was observed to be 47░ 22', and the angle of depression of its shadow S upon the sea was 61░ 10...
Page 109 - From the summit of a lighthouse 85 feet high, standing on a rock, the angle of depression of a ship was 3░ 38', and at the bottom of the lighthouse it was 2░ 43'; find the horizontal distance of the ship, and the height of the rock.

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