565 5.4 Solutions and Applications of Right Triangles 11. Find the exact value of each variable in the figure. 36 z x y w 45° 30° Find exact values of the six trigonometric functions for each angle. Rationalize denominators when applicable. 12. 135° 13. -150° 14. 1020° Find all values of u, if u is in the interval 30°, 360°2 and has the given function value. 15. sin u = 23 2 16. sec u = -22 Use a calculator to approximate the value of each expression. Give answers to six decimal places. 17. sin 42° 18′ 18. sec1-212° 12′2 Find a value of u in the interval 30°, 90°2 that satisfies each statement. Write each answer in decimal degrees to six decimal places. 19. tan u = 2.6743210 20. csc u = 2.3861147 Historical Background The beginnings of trigonometry can be traced back to antiquity. Figure 42 shows the Babylonian tablet Plimpton 322, which provides a table of secant values. The Greek mathematicians Hipparchus and Claudius Ptolemy developed a table of chords, which gives values of sines of angles between 0° and 90° in increments of 15 minutes. Until the advent of scientific calculators in the late 20th century, tables were used to find function values. Applications of spherical trigonometry accompanied the study of astronomy for these ancient civilizations. Until the mid-20th century, spherical trigonometry was studied in undergraduate courses. See Figure 43. An introduction to applications of the plane trigonometry studied in this text involves applying the ratios to sides of objects that take the shape of right triangles. 5.4 Solutions and Applications of Right Triangles ■ Historical Background ■ Significant Digits ■ Solving Triangles ■ Angles of Elevation or Depression ■ Bearing ■ Further Applications Plimpton 322 Figure 42 Figure 43
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