Find the value of x. Round to the nearest tenth, if necessary. Find the measure of angle θ. Round to the nearest degree, if necessary. 1. An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the length of the side opposite. 3. Because the length of the side opposite and adjacent to θ are given, use the tangent function. 2. An acute angle measure and the length of the side opposite the angle are given, so the tangent function can be used to find the length of the side adjacent to the angle. 4. Because the length of the side opposite and the hypotenuse are given, use the cosine function. Write each degree measure in radians as a multiple of π and each radian measure in degrees. 6. 200 To convert a degree measure to radians, multiply by esolutions Manual - Powered by Cognero Page 1
7. To convert a radian measure to degrees, multiply by Sketch each angle. Then find its reference angle. 9. 165 The terminal side of 240 lies in Quadrant II. Therefore, its reference angle is ' = 180 165 or 15. 8. Find the area of the sector of the circle shown. 10. The area of a sector A = r 2, where r is the radius and θ is the central angle. The terminal side of lies in Quadrant IV. Therefore, its reference angle is ' =. The area of the sector is about 209.4 square inches. esolutions Manual - Powered by Cognero Page 2
Find the exact value of each expression. 11. sec Because the terminal side of θ lies in Quadrant III, the reference angle θ of is. 13. MULTIPLE CHOICE An angle satisfies the following inequalities: csc θ < 0, cot θ > 0, and sec θ < 0. In which quadrant does lie? F I G II H III J IV If csc < 0, then sin < 0. So, must lie in the 3rd or 4th quadrant. If sec < 0, then cos < 0. With this additional restriction, must lie in the 3rd quadrant. With sin < 0 and cos θ < 0, cot must be > 0. The correct choice is H. Find all solutions for the given triangle, if possible. If no solution exists, write no solution. Round side lengths to the nearest tenth and angle measurements to the nearest degree. 19. a = 8, b = 16, A = 22 12. cos ( 240 ) cos ( 240 ) =cos ( 240 +360) = cos (120 ) Because the terminal side of lies in Quadrant II, the reference angle ' is. Draw a diagram of a triangle with the given dimensions. Notice that A is acute and a < b because 8< 16. Therefore, two solutions may exist. Find h. 8 > 6, so two solutions exist. Apply the Law of Sines to find B. esolutions Manual - Powered by Cognero Page 3
20. a = 9, b = 7, A = 84 Draw a diagram of a triangle with the given dimensions. Because two angles are now known, C 180 (22 + 49 ) or about 109. Apply the Law of Sines to find c. Notice that A is acute and a > b because 9 > 7. Therefore, one solution exists. Apply the Law of Sines to find B. When B 131, then C 180 (22 + 131 ) or about 27. Apply the Law of Sines to find c. Because two angles are now known, C 180 (84 + 51 ) or about 45. Apply the Law of Sines to find c. Therefore, the remaining measures of are B 49, C 109, c 20.1 and B 131, C 27, c 9.7. Therefore, the remaining measures of B 51, C 45, and c 6.4. are esolutions Manual - Powered by Cognero Page 4
21. a = 3, b = 5, c = 7 Use the Law of Cosines to find an angle 22. a = 8, b = 10, C = 46 Use the Law of Cosines to find the missing side Use the Law of Sines to find a missing angle Use the Law of Sines to find a missing angle Find the measure of the remaining angle. Find the measure of the remaining angle. Therefore, c 7.3, A 52, and B 82. Therefore, A 22, B 38, and C 120. esolutions Manual - Powered by Cognero Page 5