Sinusoidal Functions - Textbook 1 [Q&A]

CHECK Your Understanding 1. Using a graphing calculator in DEGREE mode, graph each sinusoidal function. Use the WINDOW settings shown. From the graph, state the amplitude, period, and equation of the axis for cach. a) y = 3 sin(2x) + 1/ b) y = 4 cos (0.5x) - 2 2. a) Ifb(x) = sin(5x) - 1, calculate /(25°). b) Iff(x) = cos xand f(x) = 0, list the values of x' where 0° 5 x ≤ 360°. 3. A buoy rises and falls as it rides the waves. The equation b(t) = cos (36s) ° models the displacement of the buoy, b(e), in metres at & seconds. a) Graph the displacement from 0 s to 20s, in 2.5 s intervals. b) Determine the period of the function from the graph. c) d) What is the displacement at 35 s? At what time, to the nearest second, does the displacement first reach - 0.8 m? 4. Determine the coordinates of the new point after a rotation of 50° about (0, 0) from the point (2, 0). PRACTISING 5. Using a graphing calculator and the WINDOW settings shown, graph cach function. Use DEGREE mode. Stare whether the resulting functions are periodic. If so, state whether they are sinusoidal. a) y = 3sin x + 1 c) y = cos 2x - sin x b) y = (0.004x)sin x d) y = 0.005x + sin x e) y = 0.5 cos x - 1| f y = sin 90° 6. Based on your observations in question 5, what can you conclude about any function that possesses sine or cosine in its equation? 7. If g(x) = sin x and b(x) = cos x; where 0° S x $ 360°, calculate each and explain what it means. a) 8(90°) 8. Using a graphing calculator in DEGREE mode, graph each sinusoidal function. Use the WINDOW settings shown. From the graph, state the amplitude, period, increasing intervals, decreasing intervals, and equation of the axis for each. a) y = 2 sin x + 3 b) y = 3 sinx + 1 c) y = sin (0.5x) + 2 e) y = 2 sin (0.25x) d) v = sin (2x) f) = 3 sin(0 5) ÷ 2 9. a) If f(x) = cos x, calculate f (35°). b) If g(x) = sin (2x), calculate g (10°). c) Ifh(x) = cos (3x) + 1, calculate b(20°). d) Iff(x) = cos x and f(x) = - 1, calculate x for 0° 5 x ≤ 360°. e) If f(x) = sin x and f(x) = - 1, calculate x for 0° ≤ x ≤ 360°. 10. Determine all values where sin x = cos x for - 360° ≤ x ≤ 360°. 11. a) Determine the coordinates of the new point after a rotation of 25° about (0, 0) from the point (1, 0). b) Determine the coordinates of the new point after a rotation of 80° about (0, 0) from the point (5, 0). c) Determine the coordinates of the new point after a rotation of 120° about (0, 0) from the point (4, 0). d) Determine the coordinates of the new point after a rotation of 230° about (0, 0) from the point (3, 0). 12. Sketch the sinusoidal graphs that satisfy the properties in the table. 13. Jim is riding a Ferris wheel, where z is time in seconds. Explain what each of A the following represents. a) h(10), where /(r) = 5 cos (18t)° b) b(10), where h(r) = 5 sin (18t)° 14. Compare the graphs for y = sin x and y = cos x; where 0° 5 x ≤ 360°. How are they the same, and how are they different? Extending 15. If the water level in the original water wheel situation was lowered so that three-quarters of the wheel was exposed, determine the equation of the sinusoidal function that describes the height of the nail in terms of the rotation. 16. A spring bounces up and down according to the model d(c) = 0.5 cos (120r) ° where d t) is the displacement in centimetres from the rest position and ris time in seconds. The model does not consider the effects of gravity. a) Make a table for 0 ≤ t ≤ 9. Use 0.5 s intervals. b) Draw the graph. c) Explain why the function models periodic behaviour. d) What is the relationship between the amplitude of the function and the displacement of the spring from its rest position? 6.2 Investigating the Properties of Sinusoidal Functions, pp. 363-364 1. a) The amplitude is half the distance between the maximum and minimum values. Since the minimum is - 2 and the maximum is 4, the Calculated using the formula (max - min) / 2. The period is the change in x for one complete cycle. The equation of the axis is found by averaging the maximum and minimum values. (a): If h(x) = sin(5x) - 1, then h(25°) = sin(5 * 25°) - 1. (b): If f(x) = cos x and f(x) = 0, then cos x = 0. A table is created with values for x and h(t). The function is periodic because its graph repeats at a regular interval. The function is sinusoidal because its graph looks like smooth symmetrical repeating waves. b) The function is not periodic because its graph does not repeat at a regular interval. The func- tion is not sinusoidal because its graph does not look like smooth symmetrical repeating waves. b) The period is the time it takes the graph to go through one complete cycle. Since the graph goes through one complete cycle between 1 = 0 and 1 = 10. for example, the period is 10 - 0 or 10 s. c) Since the displacement at 20 s is the same as the displacement at 0 s, the displacement at 20s + 15 s or 35 s will be the same as the dis- placement at 0s + 15 s or 15 s. The displace- ment at 15 s was - 1 m, so the displacement at 35 s will also be - 1 m. d) From the graph, the displacement first reaches - 0.8 m at about 4 s. 4. Consider the point (0, 0) the centre of a circle with radius 2, with the point (2, 0) lying on the circle. Any point on a circle centred at (0, 0) with radius r and rotated through an angle 0 can be expressed as an ordered pair (r cos 0, r sin 0). Therefore, the coordinates of the new point after a rotation of 50° about (0, 0) from the point (2, 0) are (2 cos 50°, 2 sin 50º) or (1.29, 1.53). 5. a) The function is periodic because its graph repeats at a regular interval. The function is not sinusoidal because its graph does not look like smooth symmetrical repeating waves. The function is not periodic because its graph does not repeat at a regular interval. The func- tion is not sinusoidal because its graph does not look like smooth symmetrical repeating waves. The function is periodic because its graph repeats at a regular interval. The function is sinusoidal because its graph looks like smooth symmetrical repeating waves. The function is not periodic because its graph does not repeat at a regular interval. The func- tion is not sinusoidal because its graph does not look like smooth symmetrical repeating waves. 6. It cannot necessarily be concluded that a function that possesses sine or cosine in its equation is periodic or sinusoidal. In other words, nothing can be concluded. 7. a) If g(x) = sin x, then g(90°) = sin 90° = 1. This means that when x = 90°, y = 1, or that the sine (-coordinate of a point on the unit circle) of 90° is 1. b) If h(x) = cos x, then h (90°) = cos 90° = 0. This means that when x = 90°, y = 0, or that the cosine (x-coordinate of a point on the unit circle) of 90° is 0. The amplitude is half the distance between the maximum and minimum values. Since the mini- mum is 1 and the maximum is 5, the amplitude 5 - is or 2. The period is the change in x that 2 occurs as the function goes through one com- plete cycle. Since the function goes through one complete cycle as x changes from 0° to 360°, for example, the period of the function is 360 - 0, or 360°. It's apparent from the graph that the increasing intervals are 0° to 90°, 270° 10 450°, 630° to 810°, 990° to 1170°, and 1350° to 1440°. It's apparent from the graph that the decreasing intervals are 90° to 270°, 450° to 630°, 810° to 990°, and 1170° to 1350°. The equation of the axis is the equation of the hori- zontal line that is halfway between the maxi- mum and the minimum. Since the minimum is 1and the maximum is 5. the equation of the (5+1)/2 axis is y = - or y = 3. The amplitude is half the distance between the maximum and minimum values. Since the mini- mum is - 2 and the maximum is 4, the amplitude is * - (-2) or 3. The period is the change in x that occurs as the function goes through one complete cycle. Since the function goes through one complete cycle as x changes