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Side 4 - 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...
Side 73 - 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.
Side 76 - С = с а (s — с)  Since the sine of an angle and the sine of its supplement are the same (v. ), whenever all that is given concerning an angle is the value of its sine, the angle may have either of two supplementary values. The ambiguity thus arising in the use of  is, however, removed by the consideration, that, since A, B, and C, being angles of a triangle, are each less than 180°, \ A, ^B, and •Ł С are each less than 90°.
Side ix - 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.
Side 73 - The sum of any two sides of a triangle is to their difference, as the tangent of half the sum of the angles opposite to those sides, to the tangent of half their difference.
Side 15 - 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.
Side 93 - 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'.
Side 82 - Example II. Given a = 0.3578, B = 32° 41', C = 47° 54'. Answers. C = 47° 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.
Side 94 - An observer from a ship saw two headlands ; the first bearing NE by E., and the second NW After he had sailed NNW 10.25 miles, the first headland bore E. by N., and the second WNW Find the bearing and distance of the first headland from the second.
Side 69 - 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.