The Elements of Plane Trigonometry

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4. lappuse - The COMPLEMENT OF AN ANGLE, or arc, is the remainder obtained by subtracting the angle or arc from 90°. Thus the complement of 45° is 45°, and the complement of 31° is 59°. When an angle, or arc, is greater than 90°, its complement is negative. Thus the complement of 127° is — 37°. Since the two acute angles of a right-angled triangle are together equal to a right angle, they are complements...
73. lappuse - 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.
73. lappuse - 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.
72. lappuse - TRIANGLES. §71. The sides of any triangle are proportional to the sines of the opposite angles §72.
15. lappuse - ... greater than the third side. h. Two angles are equal when their sides are respectively perpendicular ; but we must be careful to take the sides of the respective angles in the same order, and to measure the angles in the same direction, (v. § 14.) In Fig. 21, for example, FIG.
15. lappuse - If two right triangles have an acute angle of the one equal to an acute angle of the other, the other acute angles will be equal.
93. lappuse - From a window on a level with the bottom of a steeple the angle of elevation of the steeple is 40°, and from a second window 18 feet higher the angle of elevation is 37° 30'.
82. lappuse - Example II. Given a = 0.3578, B = 32° 41', C = 47° 54'. Answers. 0 = 4:7° 54', 6=0.1959, c = 0.2691. § 85. CASE II. Given two sides and an angle opposite one of them, — a, b, and A: find c, B, and C.
93. lappuse - From a station, B, at the base of a mountain, its summit A is seen at an elevation of 60° ; after walking one mile towards the summit, up a plane making an angle of 30° with the horizon, to another station, C, the angle . .BCA is observed to be 135° : find the height of the mountain in yards.
69. lappuse - Having measured a distance of 200 feet, in a direct horizontal line, from the bottom of a steeple, the angle of elevation of its top, taken at that distance, was found to be 47° 30'; from hence it is required to find the height of the steeple.

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