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Introduction to Trigonometric Graphs The following graph shows the sine function from 0º to 360º. 1) Where does sin Θ = 0? sin _______ = 0 sin ________ = 0 sin ________ = 0 2) Where does sin Θ = 1? sin _______ = 1 3) Where does sin Θ = -1? sin _______ = -1 The sine function (and the other five trigonometric functions) are periodic. This means the graph repeats itself at certain intervals. The sine function repeats itself every 360º. The graph below shows the sine function from -720º to 720º. 4) Find the values by referring to the graph above. sin 450º = _______ sin -90º = _______ sin 720º = _______ sin -270º = ______ sin -540º = ______ sin 630º = _______ 5) In the interval [-720˚, 720˚], find all values of Θ for which each equation is true. sin Θ = 0 ________________________ sin Θ = 1 ________________________ sin Θ = -1 ________________________ 6) Graph the sine curve in the interval 0º ≤ Θ ≤ 1440º 7) Approximate each value to 3 decimal places (use your calculator, f(x) = sinx) and plot the points on the graph below. to 3 places. This will give you a more accurate graph of the sine curve. sin 10º = ________ sin 20º = _______ sin 50º = _______ sin 30º = _______ sin 60º = _______ sin 70º = ______ sin 40º = _______ sin 80º = _______ sin 90º = _______ sin 180º = _______ sin 270º = _______ sin 360º = _______ WS 2.1 Refer to the Graphs of the Six Trigonometric Functions to answer the following questions. 1. What similarity do you notice between the sine and cosine curves? 2. Cosecant and secant are reciprocals of the sine and cosine ratios, respectively. What similarities do you see between the cosecant and secant curves? 3. How do the sine and cosecant graphs compare? 4. How do the cosine and secant graphs compare? 5. How do the tangent and cotangent graphs compare? A quadrantal angle is an angle in standard position whose terminal side coincides with one of the coordinate axes. The measures of the quadrantal angles are…..-270º, -180º, -90º, 0º, 90º, 180º, 270º…. Knowing the characteristics of the basic trigonometric graphs can help you quickly determine quadrantal values of the trigonometric function. 6. The value of the sine function at –360º, -180º, 0º, 180º, 360º and all other integral multiples of 180º is: a) zero b) 1 c) –1 d) undefined 7. The value of the sine function at –270º and 90º and all other integral multiples of 360º is: a) zero b) 1 c) –1 d) undefined 8. The value of the sine function at -90º and 270º and all other integral multiples of 360º is: a) zero b) 1 c) –1 d) undefined 9. The maximum value for the sine function is _______ and occurs at _______º in the interval [0, 360] 10. The minimum value for the sine function is ______ and occurs at _______º in the interval [0, 360] 11. The value of the cosine function at –360º, 0º, 360º and all other integral multiples of 180º is: a) zero b) 1 c) –1 d) undefined 12. The value of the cosine function at –270º and 90º and all other integral multiples of 360º is: a) zero b) 1 c) –1 d) undefined 13. The value of the cosine function at -90º and 270º and all other integral multiples of 360º is: a) zero b) 1 c) –1 d) undefined 14. The maximum value for the cosine function is _______ and occurs at________º in the interval [0, 360] 15. The minimum value for the cosine function is _______ and occurs at________º in the interval [0, 360] 16. The vertical lines in the graphs of the tangent, cotangent, secant, and cosecant functions are asymptotes. What do they indicate about the values of Θ where the asymptotes cross the Θ axis? The fact that the trigonometric functions are periodic allows us to find more function values easily. The period of a trigonometric function is the number of degrees taken to complete one cycle (which is the portion of the graph of the function from one point on the graph to the point at which the graph starts repeating itself). Since the graph of the sine function repeats itself every 360º, its period is 360º. This makes it easy to find other zero, maximum, and minimum values. For example, since we know that sin 90º = 1, sin (90º 360pº) = 1 for any integral value of p. The same method can be used to find special angles of other functions. 17. State the period of each of the trigonometric functions. (How many degrees does it take for the graph to complete one cycle?) sine ______________ cosecant ___________ cosine _____________ secant ___________ tangent ____________ cotangent ___________ 18. Find the values of Θ for which each equation is true. [0, 360] sin Θ = -1 cos Θ = 1 tan Θ = 0 csc Θ = undefined 19. Complete the table by finding the values of cosine Θ in the interval –540º≤ Θ ≤ 0º Plot the ordered pairs. Connect these points with a smooth continuous curve. Θ -540º -450º -360º -270º -180º -90º 0º cos Θ Find all values in the domain: [-360º, 360º] 20. cos Θ = 0 21. cos Θ = 1 22. cos Θ = -1 23. tan Θ = 0 24. tan Θ = undefined 25. csc Θ = 1 26. csc Θ = -1 27. csc Θ = undefined 28. sec Θ = 1 29. sec Θ = -1 30.. sec Θ = undefined 31. cot Θ = 0 Find each value. 32. sin (-720º) 33. cos (540º) 34. tan (-180º) 35. csc (720º) 36. sec (180º) 37. cot (180º) 38. sin (270º) 39. cos (-180º) 40. tan (270º) 41. csc (-90º) 42. sec (270º) 43. cot (-270º) Graph each function on the given interval. 44. y = sin x; -90º ≤ Θ ≤ 90º 45. y = cos x; -180º ≤ Θ ≤ 180º 46. y = tan x, -90º ≤ Θ ≤ 270º 48. y = csc x, 0º ≤ Θ ≤ 360º 50. y = sin x, -720º ≤ Θ ≤ 720º 47. y = sec x, -270º ≤ Θ ≤ 270º 49. y = cot x, -270º ≤Θ ≤ 270º WS 2.2 Find the amplitude, period, and phase shift. You may need to put in standard form first. 8. y = -4sin(6Θ – 180º) 1. y = 4sinΘ 9. y = 7cos(Θ – 45º) 2. y = 2sin5Θ 10. y = -5cos(3Θ + 60º) 3. y = 2cosΘ 11. y = 7sin5(Θ – 90º) 4. y = 3cos½Θ 12. y = -8cos(5Θ + 180º) 5. y = 6sin4Θ 13. y = 3sin(½Θ + 60º) 6. y = 10cos6Θ 14. y = 4cos (3Θ – 30º) 7. y = 5sin(Θ + 90º) Write an equation of the sine function given the amplitude, period, and phase shift. 15. a: ¾ p: 360º ps: 30º ____________________________ 16. a: 4 p. 180º ps: -30º ____________________________ 17. a: 5 p: 45º ps: -60º ____________________________ 18. a: 7 p: 225º ps: 90º ____________________________ 19. a: 9 p: 30º ps: -180º ____________________________ 20. a: 1 p: 60º ps: 120º ____________________________ Write an equation of the cosine function given the amplitude, period, and phase shift. 21. a: 1/3 p: 180º ps: 0º ____________________________ 22. a: 3 p: 180º ps: 120º ____________________________ 23. a: 100 p: 90º ps: -45 ____________________________ 24. a: 1 p: 360º ps: -90º ____________________________ 25. a: 3.75 p: 90º ps. 4º ____________________________ 26. a: 12 p: 45º ps: -180º ____________________________ Graph each function in the domain of the graph. 27. y = ½sin x 28. y = cos(2x) 30. y = sin(½x) 33. y = 4sinx 31. y = 3cos x 29. y = sin(x – 90º) 32. y = cos(x + 90º) WS 2.3 Put in standard form, if necessary. Find the amplitude, period, phase shift, vertical shift, frequency, and DBCV. Sketch the graph of one complete cycle on notebook paper clearly showing the critical points. 1. y = 4 cos(3Θ – 30º) + 7 2. y = -2 sin(2Θ + 60º) - 10 3. y = 5 cos(½Θ + 45º) + 3 4. y = 6 sin(Θ – 30º) + 11 5. y = -2 - 3 cos(4Θ – 180º) 6. 4 + y = - 5 sin(10Θ + 300º) 7. y = 3 + 6cos(Θ – 15º) 3 Given the following graphs, write the equation of the sinusoid as a “+ sine” function” and as a “- sine” function. 8. 9. 10. WS 2.4 Sketch one cycle of the graph of the circular function. 2 4. f (x ) 2 6 sin (x 1) 1. f (x ) 4 5 sin x 4 3 2 2. f (x ) 4 3cos 1 (x ) 4 5. f (x ) 4 5 cos 3 (x 2) 6. 12 36sin 3x 4y 15 4 8cos 2x 9 3. f (x ) 2 Use the graphs shown on the back of this worksheet to write an equation of the sinusoids shown there for the following problems #7 – 12, using a circular function. 7. “+” cosine, “-” sine 10. “+” sine, “+” cosine 8. “-” cosine, “+” sine 11 “-” cosine, “-” sine 9. “+” cosine, “-” sine 12. “+” cosine, “-” sine Write the equation of the given sinusoid. 13. sine: amplitude: 12, period: 2 5 14. cosine: amplitude: 8, period: , phase shift: left , vertical shift: down 5 2 , phase shift: right , vertical shift: up 2 4 6 15. sine: period: 10π, phase shift: left π, vertical shift: down 4; amplitude: 1 16. cosine: amplitude: 12, period: 8π, phase shift: right , vertical shift up 10 4 17. sine: amplitude: 1, period : 30, phase displacement: right 4, vertical shift down 3 18. sine: amplitude: ½, period: 12, phase displacement: left 10, vertical shift down 2 19. cosine: amplitude: 7, period: 10, phase shift: left 2, vertical displacement: up 8 20. cosine: amplitude: 4, period: 6, phase shift: right 6, vertical displacement: none Graphs for Worksheet 2.4 #7-12 WS 2.5 Sketch one cycle of the given function . (1 - 4: trigonometric, 5 - 8: circular) 1. y = tan2θ 2. y = 2cscθ 3. y = 4cot3θ 4. y = cot((π/3)x) Radians 5. y = 3sec(2x) Radians 6. y = -5 + 3cot4θ 7. y = -1 + 3tan2(θ - 30°) 8. y = 3 + 2sec4(θ + 10°) 9. y = -2 + 4csc(x + 4) Radians 10. y = -3 + 6csc((π/2(x + 1) Radians