Find the value of x. Round to the nearest tenth, 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. 2. An acute angle measure and the length of the side opposite θ are given, so the tangent function can be used to find the length of the side adjacent. Find the measure of angle θ. Round to the nearest degree, if necessary. 3. Because the length of the side opposite and adjacent to θ are given, use the tangent function. esolutions Manual - Powered by Cognero Page 1
4. Because the length of the side opposite and the hypotenuse are given, use the cosine function. 5. MULTIPLE CHOICE What is the linear speed of a point rotating at an angular speed of 36 radians per second at a distance of 12 inches from the center of the rotation? A 420 in./s B 432 in./s C 439 in./s D 444 in./s The formula for linear speed is v = radius and θ is the central angle. Thus, where s is the arc length traveled. Arc length is equal to rθ, where r is the. The distance from the center of rotation, also known as the radius, is 12 inches, so. The formula for angular speed is. The angular speed is 36 radians per second, so. Therefore, The correct choice is B. esolutions Manual - Powered by Cognero Page 2
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 7. To convert a radian measure to degrees, multiply by 8. Find the area of the sector of the circle shown. The area of a sector A = r 2, where r is the radius and θ is the central angle. The area of the sector is about 209.4 square inches. esolutions Manual - Powered by Cognero Page 3
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. 10. The terminal side of lies in Quadrant IV. Therefore, its reference angle is ' =. Find the exact value of each expression. 11. sec Because the terminal side of θ lies in Quadrant IV, the reference angle ' is or. esolutions Manual - Powered by Cognero Page 4
12. cos ( 240 ) Because the terminal side of lies in Quadrant II, the reference angle ' 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 3 rd or 4 th quadrant. If sec < 0, then cos < 0. With this additional restriction, must lie in the 3 rd quadrant. With sin < 0 and cos θ < 0, cot must be > 0. The correct choice is H. esolutions Manual - Powered by Cognero Page 5
State the amplitude, period, frequency, phase shift, and vertical shift of each function. Then graph two periods of the function. 14. y = 4 cos 5 In this function, a = 4, b =, c = 0, and d = 5. Graph y = 4 cos shifted 5 units down. esolutions Manual - Powered by Cognero Page 6
15. In this function, a = 1, b = 1, c =, and d = 0. Because d = 0, there is no vertical shift. Graph y = sin x shifted units to the left. esolutions Manual - Powered by Cognero Page 7
16. TIDES The table gives the approximate times that the high and low tides occurred San Azalea Bay over a 2-day period. a. The tides can be modeled with a trigonometric function. Approximately what is the period of this function? b. The difference in height between the high and low tides is 7 feet. What is the amplitude of this function? c. Write a function that models the tides where t is measured in hours. Assume the function has no phase shift or vertical shift. a. The period lasts from peak to peak, or from high tide to high tide. 3:04 PM is 12 hours and 30 minutes after 2:35 AM. b. The amplitude is one half of the difference between the maximum (high tide) and the minimum (low tide). One half of 7 is 3.5. c. The period is 12 hours and 30 minutes, or 12.5 hours. esolutions Manual - Powered by Cognero Page 8
Locate the vertical asymptotes, and sketch the graph of each function. 17. The graph of is the graph of y = tan x translated units to the left. The period is or. Find the location of two consecutive vertical asymptotes. and Create a table listing the coordinates of key points for for one period on. Functions Vertical Asymptote Intermediate Point x-intercept Intermediate Point Vertical Asymptote y = tan x x = (0, 0) x = x = x = Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one more cycle to the left and right of the first curve. esolutions Manual - Powered by Cognero Page 9
18. y = sec (2x) The graph of y = sec 2x is the graph of y = sec x compressed horizontally and compressed vertically. The period is or π. Find the location of two vertical asymptotes. and Create a table listing the coordinates of key points for y = sec 2x for one period on. Function Vertical Asymptote Intermediate Point Vertical Asymptote Intermediate Point y = sec x x = x = x = y = sec 2x x = x = x = Vertical Asymptote Sketch the curve through the indicated key points for the function. Then repeat the pattern to sketch at least one more cycle to the left and right of the first curve. 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 Draw a diagram of a triangle with the given dimensions. esolutions Manual - Powered by Cognero Page 10
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. Because two angles are now known, C 180 (22 + 49 ) or about 109. Apply the Law of Sines to find c. When B 131, then C 180 (22 + 131 ) or about 27. 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. esolutions Manual - Powered by Cognero Page 11
20. a = 9, b = 7, A = 84 Draw a diagram of a triangle with the given dimensions. Notice that A is acute and a > b because 9 > 7. Therefore, one solution exists. Apply the Law of Sines to find B. 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 B 51, C 45, and c 6.4. are esolutions Manual - Powered by Cognero Page 12
21. a = 3, b = 5, c = 7 Use the Law of Cosines to find an angle measure. Use the Law of Sines to find a missing angle measure. Find the measure of the remaining angle. Therefore, A 22, B 38, and C 120. esolutions Manual - Powered by Cognero Page 13
22. a = 8, b = 10, C = 46 Use the Law of Cosines to find the missing side measure. Use the Law of Sines to find a missing angle measure. Find the measure of the remaining angle. Therefore, c 7.3, A 52, and B 82. Find the exact value of each expression, if it exists. 23. Find a point on the unit circle on the interval with a x-coordinate of. When t =, cos t =. Therefore, cos 1 =. esolutions Manual - Powered by Cognero Page 14
24. Find a point on the unit circle on the interval with a y-coordinate of. When t =, sin t =. Therefore, sin 1 =. esolutions Manual - Powered by Cognero Page 15
25. NAVIGATION A boat leaves a dock and travels 45º north of west averaging 30 knots for 2 hours. The boat then travels directly west averaging 40 knots for 3 hours. a. How many nautical miles is the boat from the dock after 5 hours? b. How many degrees south of east is the dock from the boat s present position? a. During the first leg of the trip, the boat traveled 40 knots for 3 hours, so the distance the boat traveled is 40 3 or 120 nautical miles. During the second leg of the trip the boat traveled 30 knots for 2 hours, and therefore traveled a distance of 30 2 or 60 nautical miles. Draw a diagram to model the situation. Let x represent the distance the boat has traveled from the dock after 5 hours. Use the Law of Cosines to find b. Therefore, the boat is 167.9 nautical miles from the dock after 5 hours. b. Use the Law of Sines to find C. Therefore, the dock is about 15 south of east from the boat's current position. esolutions Manual - Powered by Cognero Page 16