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Topic 4 Periodic Functions & Applications II 1. 2. 3. 4. 5. 6. 7. 8. Definition of a radian and its relationship with degrees Definition of a periodic function, the period and amplitude Definitions of the trigonometric functions sin, cos and tan of any angle in degrees and radians Graphs of y = sin x, y = cos x and y = tan x Significance of the constants A, B, C and D on the graphs of y = A sin(Bx + C) + D, y = A cos(Bx + C) + D Applications of periodic functions Solutions of simple trigonometric equations within a specified domain Pythagorean identity sin2x + cos2x = 1 1. Definition of a radian and its relationship to degrees Radians In the equilateral triangle, each angle is 60o r 60 r If this chord were pushed onto the circumference, this radius would be pulled back onto the other marked radius Radians 1 radian 2 radians 3 radians 57o18’ 114o36’ 171o54’ radians = 180o Radians radians = 180o /2 radians = 90o /3 radians = 60o /4 radians = 45o etc Model Express the following in degrees: (a) 2 3 (b) 4 5 (c) 13 6 Remember = 180o 2 2 180 2 60 120 3 3 4 4 180 4 36 144 5 5 13 13 180 13 30 390 6 6 Model Express the following in radians: (a) 225 (b) 72 (c) 43 Remember = 180o 225 225 180 54 72 72 180 52 43 43 180 43 180 5 4 2 5 Exercise NewQ P 298 Set 9.1 Numbers 1 - 4 2. Definition of a periodic function, period and amplitude • Consider the function shown here. • A function which repeats values in this way is called a Periodic Function • The minimum time taken for it to repeat is called the Period (T). This graph has a period of 4 • The average distance between peaks and troughs is called Amplitude (A). This graph has an amplitude of 3 3. Definition of the trigonometric functions sin, cos & tan of any angle in degrees and radians Unit Circle Model Find the exact value of: (a) cos 225 (b) tan 225 (c) sin 300 Model Find the exact value of: (a) cos 225 (b) tan 225 (c) sin 300 cos 225 cos 45 1 2 45 Model Find the exact value of: (a) cos 225 (b) tan 225 (c) sin 300 tan 225 tan 45 1 45 Model Find the exact value of: (a) cos 225 (b) tan 225 (c) sin 300 sin 300 sin 60 3 2 60 Now let’s do the same again, using radians Model Find the exact value of: (a) (b) (c) cos 225 5 tan 225 sin 300 4 5 3 Model 5 cos 4 5 (b) tan 4 5 sin (c) 3 Find the exact value of: (a) 5 cos 4 cos 1 2 4 4 Model Find the exact value of: (a) (b) (c) 5 tan 4 tan 1 4 5 cos 4 5 tan 4 5 sin 3 4 Model Find the exact value of: (a) (b) (c) 5 sin 3 sin 3 2 3 5 cos 4 5 tan 4 5 sin 3 3 Exercise NewQ P 307 Set 9.2 Numbers 1, 2, 8-11 4. Graphs of y = sin x, y = cos x and y = tan x The general shapes of the three major trigonometric graphs y = sin x y = cos x y = tan x 5. Significance of the constants A,B and D on the graphs of… y = A sinB(x + C) + D y = A cosB(x + C) + D 1. Open the file y = sin(x) y = A cos B(x + C) + D A: adjusts the amplitude B: determines the period (T). This is the distance taken to complete one cycle where T = 2/B. It therefore, also determines the number of cycles between 0 and 2. C: moves the curve left and right by a distance of –C (only when B is outside the brackets) D: shifts the curve up and down the y-axis Graph the following curves for 0 ≤ x ≤ 2 a) y = 3sin(2x) b) y = 2cos(½x) + 1 Exercise NewQ P 318 Set 9.4 1 - 6 6. Applications of periodic functions Challenge question Assume that the time between successive high tides is 4 hours High tide is 4.5 m Low tide is 0.5m It was high tide at 12 midnight Find the height of the tide at 4am Assume that the time between successive high tides is 4 hours High tide is 4.5 m Low tide is 0.5m It was high tide at 12 midnight Find the height of the tide at 4am Assume that the time between successive high tides is 4 hours High tide is 4.5 m Low tide is 0.5m It was high tide at 12 midnight y = a sin b(x+c) + d Tide range = 4m a = 2 Find the height of the tide at 4am High tide = 4.5 d = 2.5 y Period = 4 b = 0.5 Period = 2/b 4 3 2 1 x 0 Assume that the time between successive high tides is 4 hours High tide is 4.5 m Low tide is 0.5m It was high tide at 12 midnight y = 2 sin 0.5(x+c) + 2.5 Find the height of the tide at 4am At the moment, high tide is at hours y We need a phase shift of units to the left 4 3 c= 2 1 x 0 Assume that the time between successive high tides is 4 hours High tide is 4.5 m Low tide is 0.5m It was high tide at 12 midnight y = 2 sin 0.5(x+) + 2.5 Find the height of the tide at 4am We want the height of the tide when t = 4 On GC, use 2nd Calc, value y 4 3 h= 1.667m 2 1 x 0 Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time: (a) Find the period and amplitude of the movement. (b) Predict the displacement at 10 seconds. (c) Find all the times up to 20 sec when the displacement will be 5 cm to the right (shown as positive on the graph) Y 8 6 4 2 0 -2 -4 -6 -8 X 1 2 3 4 5 Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time: (a) Find the period and amplitude of the movement. (b) Predict the displacement at 10 seconds. (c) Find all the times up to 20 sec when the displacement will be 5 cm to the right (shown as positive on the graph) Period = 4.5 - 0.5 Y = 4 sec 8 6 4 2 0 -2 -4 -6 -8 X 1 2 3 4 5 Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time: (a) Find the period and amplitude of the movement. (b) Predict the displacement at 10 seconds. (c) Find all the times up to 20 sec when the displacement will be 5 cm to the right (shown as positive on the graph) Amplitude = 8 Y 8 6 4 2 0 -2 -4 -6 -8 X 1 2 3 4 5 Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time: (a) Find the period and amplitude of the movement. (b) Predict the displacement at 10 seconds. (c) Find all the times up to 20 sec when the displacement will be 5 cm to the right (shown as positive on the graph) Since the period = 4 sec Y 8 Displacement after 10 sec should be the same as displacement after 2 sec 6 4 2 0 -2 X 1 -4 -6 -8 = 5.7cm to the left 2 3 4 5 Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time: (a) Find the period and amplitude of the movement. (b) Predict the displacement at 10 seconds. (c) Find all the times up to 20 sec when the displacement will be 5 cm to the right (shown as positive on the graph) Displacement= 5cm Y 8 t= 6 4 1.1 5.1, 9.1, 13.1, 17.1 3.9 7.9, 11.9, 15.9, 19.9 2 0 -2 -4 -6 -8 X 1 2 3 4 5 Exercise NewQ P 179 Set 5.2 1,3 Model: Find the equation of the curve below. y = a sin b(x+c) Amplitude = 2.5 Y 2 X 0 -2 1 2 3 4 5 6 7 8 9 10 Model: Find the equation of the curve below. y = 2.5 sin b(x+c) Amplitude = 2.5 Period = 6 6 = 2/b Period = 2/b b = /3 Y 2 X 0 -2 1 2 3 4 5 6 7 8 9 10 Model: Find the equation of the curve below. Amplitude = 2.5 Period = 6 6 = 2/b Period = 2/b y = 2.5 sin /3(x+c) Phase shift = 4 () so c = -4 b = /3 Y 2 X 0 -2 1 2 3 4 5 6 7 8 9 10 Model: Find the equation of the curve below. Amplitude = 2.5 Period = 6 6 = 2/b Period = 2/b y = 2.5 sin /3(x-4) Phase shift = 4 () so c = -4 b = /3 Y 2 X 0 -2 1 2 3 4 5 6 7 8 9 10 Exercise NewQ P 183 Set 5.3 1,4 Find the equation of the curve below in terms of the sin function and the cosine function.