Elements of Trigonometry, Plane and Spherical

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Sheldon, 1872 - Trigonometry - 201 pages
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Page 50 - C' (89) (90) (91) (92) (93) 112. 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.
Page 5 - The logarithm of a quotient equals the logarithm of the dividend minus the logarithm of the divisor.
Page 96 - In a spherical triangle the cosine of any side is equal to the product of the cosines of the other two sides plus the product of the sines of these two sides and the cosine of their included angle.
Page 62 - Parallels, which are imaginary circles parallel with the equator, determine latitude. The length of a degree of longitude varies as the cosine of the latitude. At the equator a degree is 69.171 statute miles; this is gradually reduced toward the poles.
Page 2 - Whence xz is the logarithm of the zth power of m. QE D 181. Prop. 4. — The logarithm of any root of a number is the logarithm of the number divided by the number expressing the degree of the root. DEM. — Let a be the base, and x the logarithm of m. Then ar=m. Extracting the th root we have a"= ^/m.
Page 49 - Ans. ^. 7. Prove that the sides of any plane triangle are proportional to the sines of the angles opposite to these sides. If 2s = the sum of the three sides (a, b, c) of a triangle, and if A be the angle opposite to the side a, prove that o sin A = T- Vs (s — a) (s — b) (s — c).
Page 74 - If the two sides of a right angled spherical triangle about the right angle be of the same affection, the hypotenuse will be less than a quadrant ; and if they be of different affection, the hypotenuse will be greater than a quadrant. Let ABC be a right angled spherical triangle ; according as the two sides AB, AC are of the same or of different affection, the hypotenuse BC...
Page 87 - ... perpendicular falls without the triangle. In this case the shorter segment lies in an opposite direction from its angle to that considered in the demonstration, and hence is to be considered — ; and s + a' is in every case equal to the side upon which the perpendicular is let flL 135.
Page 4 - ... being taken as to whether it is integral, fractional, or mixed ; as in any case, the figures will be the вате).
Page 77 - The sine of the middle part equals the product of the tangents of the adjacent parts.

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