Plane and Spherical Trigonometry

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Ginn, Heath & Company, 1882 - Trigonometry - 201 pages

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Page 126 - 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 77 - I. The sine of the middle part is equal to the product of the tangents of the adjacent parts.
Page 20 - Geometry that the area of a triangle is equal to one-half the product of the base by the altitude. Therefore, if a and b denote the legs of a right triangle, and F the area...
Page 52 - In any triangle, the square of a side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the projection of the other side upon it.
Page 23 - From the top of a hill the angles of depression of two objects situated in the...
Page 52 - The square of any side of a triangle is equal to the sum of the squares of the other two sides, diminished by twice the product of the sides and the cosine of the included angle.
Page 108 - ZOB between the zenith of the observer and the celestial equator is obviously equal to his latitude, and the angle POZ is the complement of ZOB. The arc NP being the complement of PZ, it follows that the altitude of the elevated pole is equal to the latitude of the place of observation. The triangle ZPM then (however much it may vary in shape for different positions of the star M), always contains the following five magnitudes : PZ= co-latitude of observer = 90°...
Page 130 - Express in degrees, minutes, etc., (i.) the angle whose circular measure is .jVir; (ii.) the angle whose circular measure is 5. If the angle subtended at the centre of a circle by the side of a regular pentagon be 'the unit of angular measurement, by what number is a right angle represented? 2. Find, by geometrical constructions, the cosine of 45° and the sine of 120°. Prove that (sin 30° + cos 30°) (sin 120° + cos 120°) = sin 30°.
Page 104 - The vertical circle passing through the east and west points of the horizon is called the Prime Vertical; that passing through the north and south points coincides with the celestial meridian.
Page 55 - Two observers 15 miles apart on a plain, and facing each other, find that the angles of elevation of a balloon in the same vertical plane with themselves are 55° and 58°, respectively.

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