Unit 4 Example Review. 1. Question: a. [A] b. [B] c. [C] d. [D]

Transcription

1 Unit 4 Example Review 1. Question: Answer: When the line is wrapped around the circle, the length of the line will be equal to the length of the arc that is formed by the wrap starting from the point (1,0) to the final point where this line would end. Length of the arc = (theta)*r = 4.44 units Radius (R) : 1 unit ð Theta = Length of the arc / R = 4.44/1 = 4.44

2 2. Question:

3 3. Question: Answer: Sin (theta) = Opposite side / Hypotenuse

4 4. Question: Answer: Upon substituting x = (2*pi), we get 0 as seen in the graph only with option C Using a calculator, the values for the above functions at x = 2*pi could be found.! 5. Change 120 to radian measure. a. b. c. d.

5 Answer: Radian measure = (pi/180) * Degree measure We know that (2*pi) = 360 degrees. => 120 degress = (360/3) degrees = (2*pi)/3 Radians 6. Question: Answer: Csc(Theta) is the multiplicative inverse (reciprocal) of sin(theta) 7. Question: Answer: The ladder s length falls along the hypotenuse, The angle 62 degree is made by the hypotenuse with the base (ground), The topmost point of the ladder touches the wall at a height that equals to the opposite of the triangle. Sin(theta) = opp / hypotenuse => height / ladder length = sin (62) => height = ladder length * sin(62) => height = 20 * (0.883) = ft

6 8. What is the degree measure of the angle through which the hour hand on a clock rotates from 4:00 P.M. to 8:00 P.M.? Answer: 12 hours make one complete revolution, i.e,360 degrees, that would imply 4 hours (4pm to 8pm) would make Number of hours that is equal to (360/12) * 4 = 360/3 = 120 degrees a. 120 b. -90 c. 90 d Question: Answer: Use a calculator

7 10. Question: Answer: Amplitude is the maximum absolute value that the motion can reach to. Here the value of the sine function is limited to one, which makes 5 as the amplitude. Time period is the time that is taken to complete one complete motion. Here, it would mean that it is equal to the time that is taken to complete (2*pi) radians. When t = (pi/2), the argument inside sine is equal to (4 * t) = (4* (pi/2)) = (2*pi) ð t = pi/ 2 and amplitude = 5cm is the solution

8 11. Question: Answer: Adding any multiple of the period of a sinusoidal function to the argument would never change the value of the function. Here, 6*pi is a multiple of 2*pi 12. Question: Answer: As explained in one of the previous problems Radian measure = (pi/180) * Degree measure Radian_measure = (pi/180) * 18 = pi/10

9 13. Question: Answer: When x goes to infinity, the given function s value goes to zero because of a very high negative exponent. As x goes to negative infinity, the functions value goes to infinity because of very high positive exponent. Only graph D satisfies the criteria

10 14. Question: Answer: Substituting x = 0, Options A and D would give infinity which clearly contradicts the graph. Substituting x = (2*pi) makes option B go to ) which again rules that out as the graph says different. The only option that worked so far is C

11 15. Question: Answer: An even function is a function that remains unchanged when the value of the input variable is changed signs. That is, y = f(x) = f(-x) The mod function (Absolute value function) returns a positive number independent of the sign of the input number. Also cosine functions is independent of the sign of the argument, cos(-x) = cos(x) The given function is an even function, that should make the graphs symmetric about y axis. Also there is a cosine function which makes the graph take both sides of the x-axis, Hence, option D

12 16. Use the unit circle to find the value of cos 120. Answer: In an unit circle, 120 degree would lie on the same level of y-axis as the 60 degree but in second quadrant. That makes it negative to the value of cos(60) a. b. c. -2 d. 17. Question: Answer: Use a calculator 18. is an acute angle of a right triangle. If sin find cos and tan. a. cos and tan

13 b. cos and tan c. cos and tan d. cos and tan Question Incomplete 19. Question:Use a calculator to approximate the expression. Answer: Use a calculator 20. Question: Answer: He is just headed 23 degrees to south which obviously means he travelled more along west and less along south. Also in one hour he must have travelled (cos(23) * 3.9) miles along west. In 3.5 hours, he would have travelled (3.9*3.5*cos(23)) = 12.6 miles

14 21. Question: Answer: Converting minutes and seconds to degrees, 68 + (13/60) + (49/3600) = Csc( ) is C 22. What is the reference angle of 240? Answer: 240 = When it comes to arguments of sinusoidal functions, it doesn t matter whether the the argument if summed up or decremented by the value of the period => Ref angle = = -120 a. 60 b. 60 c. 120 d Question:Use the properties of inverse functions to evaluate the expression.

15 Answer: Simply put, the task is to find the cosine of the angle whose sine value is (2*(x^0.5))/(1+x) Sin(theta) = height / hypotenuse = (2*(x^0.5))/(1+x) Height = (2*(x^0.5)) Hypotenuse = (1+x) Base = (Hypotenuse^2 - Height^2)^0.5 = ((1+x)^2 4*x)^0.5 = ((1-x)^2)^0.5 = (1-x) Cos(theta) = base/ hypotenuse = (1-x)/(1+x) 24. Question: Answer: 1 revolution = 2*pi radians ð 0.7 revolutions = 0.7 * 2 * pi radians = (7*pi)/5 radians

16 25. Find the value of csc by referring to the graph of the cosecant function. Answer: The graph isn t attached a. -1 b. 0 c. 1 d. undefined 26. Use the unit circle to find sec (-180 ). a. -1 b. 0 c. undefined d. 1 Answer: -180 degree would lie at the same point as +180 degree, which is symmetric to the point at 0 degree with respect to y axis. => Answer = -sec(0) = Find sin, cos, and tan. Answer: Sin(theta) = y / (y^2 + x^2) = -(0.5)/(1) = -0.5 Cos(theta) = x / (y^2 + x^2) = -(3^0.5)/2

17 Tan(theta) = sin(theta)/tan(theta) = 1/(3^0.5) a. b. c. d. 28. Question: Answer: cos(theta) = base/ hypotenuse = b/c

18 29. Question: Answer: Use calculator

19 30. Question: Answer: Tracing negative(7*pi)/6, reach to the first point above the x axis in the second quadrant which is same as cot(5*pi/6). By symmetry, it is negative of the value that is in first quadrant at the same y-axis level. => cot(-(7*pi)/6) = -cot(pi/6) => -cot(pi/6) = -1/(3^0.5) => Option D

20 31. Question: Answer: phase shift is the addition to the argument and tie period is the time at which the sinusoidal function would reach 2*pi in absence of the phase shift Here, -(5*pi)/4 is the phase shift Let time period = T (3 * T)/4 = 2*pi ð T = 8*pi/3 32. Find the exact value of. a. b. c. d. -2

21 Answer: 2*pi/3 is symmetric about y axis to the angle pi/3, secant is negative in second quadrant. Hence answer = -sec(pi/3) = State the amplitude for the function y = - cos. Answer: Amplitude is the absolute maximum value of the given function, maximum value of cosine function is 1, Hence max value is absolute(-2/3) = 2/3 a. - b. c. 1 d Suppose Alex wants to kick a ball over a 10-foot fence. Assuming that gravity has no effect, at what angle would he have to kick the ball if he were standing 6 feet away from the fence? Answer: Height: 10 foot Base = 6 feet Tan(theta) = height / base = 10/6 ð theta = atan(10/6) a b c d State the amplitude, period, phase shift, and vertical shift for (Incomplete)

22 a. b. c. d. 36. Question: Answer: Rises two feet for 5 feet horizontal ð tan(theta) = 2/5 ð 1320/b = 2/5 ð b = (5/2) * 1320 = 3300 ð c = (b^ ^2)^0.5 =

23 37. Find the exact value of sin, cos, and tan if the terminal side of in standard position contains the point (5, 12). Answer: Tan(theta) = 12/5 Cos(theta) = 5/13 Sin(theta) = 12/13 a. sin, cos, tan b. sin, cos, tan c. sin, cos, tan d. sin, cos, tan

24 38. Questions:

25 39. Question: Answer: arcsin(0.5) = 30 degree Tan(30) = 1/(3^0.5) Answer: 40. Question: Answer: Theta_ dash = theta + 2*pi = 2*pi (9*pi/5) = (10*pi 9*pi)/5 = pi/5; pi/5 lies in the first quadrant

26 41. Write the equation for the inverse of y = Arctan 4x. Answer: y = atan(4x) ð tan(y) = 4x ð x = 0.25 * tan(y) ð Reqriting the variables, y = (1/4)*tan(y) a. y = tan x b. y = 2 tan x c. y = 4 tan x d. y = tan x 42. Question: Answer: Use calculator 43. Find one positive angle and one negative angle that are coterminal with 310. Answer: Positive angle = = 370;

27 Negative angle = = -50 a. 100, -670 b. 50, -670 c. -120, 400 d. -50, Determine if the function is periodic. If so, state the period. Answer: Yes, It is periodic as it repeats itself after a specific number of units in x-axis. If noted clearly, the triangle is the repeating unit and it repeats every 2 units of x-axis. a. no b. yes; 4 c. yes; 2 d. yes; Question:

28 Answer: Height (Constant) = 30.5 km Base 1 = height / tan(63.6) = 15, Base2 = height/tan(56.1) = 20, Distance covered = 20, = 5, Speed = Distance / Time = 5, /33 = m/s 46. Question: Answer: Period = 2*pi Initial argument = (-63*pi)/8 Adding (4*2*pi) to it We get, 8*pi + (-63*pi)/8 = (64-63)*(pi/8) = pi/8

29 47. How is the function y = 3 sin ( +30) shifted? Answer: Let the function without the phase shift be z = 3*sin(theta) If theta = 0, z = 0 Now consider y = 3 * sin(theta+30) For the argument inside sine to be zero, theta + 30 = 0 ð Theta = -30 Theta is equal to -30 to get the same value for the function, Hence the graph would be shifted left. a. shifted up 30 b. shifted 30 to the right c. shifted 30 to the left d. shifted down Question: Answer: cot(beta) * sin(beta)

30 = (cos(beta)/sin(beta)) * sin(beta) = cos(beta) * (sin(beta)/ sin(beta)) = cos(beta) 49. Question: 50. Determine the exact value of sin 150. a. b. c. d. Answer: 150 degree is symmetric about y axis to the angle 30 degree. Sine is positive in the second quadrant

31 ð Sin(150) = sin(30) = 1/2