786 CHAPTER 8 Applications of Trigonometry Solve each problem. 83. Area of a Metal Plate A painter is going to apply paint to a triangular metal plate on a new building. Two sides measure 16.1 m and 15.2 m, and the angle between the sides is 125°. What is the area of the surface she plans to paint? 84. Area of a Triangular Lot A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides are 52.1 m and 21.3 m, and the angle between them is 42.2°. What is the area of the triangular lot? 85. Triangle Inscribed in a Circle For a triangle inscribed in a circle of radius r, the law of sines ratios a sin A , b sin B , and c sin C have value 2r. The circle in the figure has diameter 1. What are the values of a, b, and c? (Note: This result provides an alternative way to define the sine function for angles between 0° and 180°. It was used nearly 2000 yr ago by the mathematician Ptolemy to construct one of the earliest trigonometric tables.) 86. Theorem of Ptolemy The following theorem is attributed to Ptolemy: In a quadrilateral inscribed in a circle, the product of the diagonals is equal to the sum of the products of the opposite sides. (Data from Eves, H., An Introduction to the History of Mathematics, Sixth Edition, Saunders College Publishing.) The circle in the figure has diameter 1. Use Ptolemy’s theorem to derive the formula for the sine of the sum of two angles. 87. Law of Sines Several of the exercises on right triangle applications involved a figure similar to the one shown here, in which angles a and b and the length of line segment AB are known, and the length of side CD is to be determined. Use the law of sines to obtain x in terms of a, b, and d. 88. Aerial Photography Aerial photographs can be used to provide coordinates of ordered pairs to determine distances on the ground. Suppose we assign coordinates as shown in the figure. If an object’s photographic coordinates are 1x, y2, then its ground coordinates 1X, Y2 in feet can be computed using the following formulas. X = 1a - h2x ƒ sec u - y sin u , Y = 1a - h2y cos u ƒ sec u - y sin u Here, f is focal length of the camera in inches, a is altitude in feet of the airplane, and h is elevation in feet of the object. Suppose that a house has photographic coordinates 1xH, yH2 = 10.9, 3.52 with elevation 150 ft, and a nearby forest fire has photographic coordinates 1xF, yF2 = 12.1, -2.42 and is at elevation 690 ft. Also suppose the photograph was taken at 7400 ft by a camera with focal length 6 in. and tilt angle u = 4.1°. (Data from Moffitt, F., and E. Mikhail, Photogrammetry, Third Edition, Harper & Row.) (a) Use the formulas to find the ground coordinates of the house and the fire to the nearest tenth of a foot. (b) Use the distance formula d = 21x2 - x122 + 1y 2 - y122 to find the distance on the ground between the house and the fire to the nearest tenth of a foot. A B C c b a A B cos A sin A sin B cos B sin (A + B) 1 C D a b B d x A (xF, yF) (xH, yH) (XF, YF) (XH, YH) u
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