787 8.2 The Law of Cosines Relating Concepts For individual or collaborative investigation (Exercises 89 – 92) Colors of the U.S. Flag The flag of the United States includes the colors red, white, and blue. Which color is predominant? Clearly the answer is either red or white. (It can be shown that only 18.73% of the total area is blue.) (Data from Banks, R., Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press.) To answer this question, work Exercises 89–92 in order. 89. Let R denote the radius of the circumscribing circle of a five-pointed star appearing on the American flag. The star can be decomposed into ten congruent triangles. In the figure, r is the radius of the circumscribing circle of the pentagon in the interior of the star. Show that the area of a star is = c 5 sin A sin B sin1A + B2 d R2 . (Hint: sin C = sin3180° - 1A + B24 = sin1A + B2.) R r A B R r C 90. Angles A and B have values 18° and 36°, respectively. Express the area of a star in terms of its radius, R. 91. To determine whether red or white is predominant, we must know the measurements of the flag. Consider a flag of width 10 in., length 19 in., length of each upper stripe 11.4 in., and radius R of the circumscribing circle of each star 0.308 in. The thirteen stripes consist of six matching pairs of red and white stripes and one additional red, upper stripe. Therefore, we must compare the area of a red, upper stripe with the total area of the 50 white stars. (a) Compute the area of the red, upper stripe. (b) Compute the total area of the 50 white stars. 92. Which color occupies the greatest area on the flag? If we are given two sides and the included angle (Case 3) or three sides (Case 4) of a triangle, then a unique triangle is determined. These are the SAS and SSS cases, respectively. Both require using the law of cosines to solve the triangle. The following property is important when applying the law of cosines. 8.2 The Law of Cosines ■ Derivation of the Law of Cosines ■ Using the Law of Cosines ■ Heron’s Formula for the Area of a Triangle ■ Derivation of Heron’s Formula Triangle Side Length Restriction In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
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