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Topic 4 Periodic Functions & Applications II 1.Definition of a radian and its relationship with degrees 2.Definition of a periodic function, the period and amplitude 3.Definitions of the trigonometric functions sin, cos and tan of any angle in degrees and radians 4.Graphs of y = sin x, y = cos x and y = tan x 5.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 6.Applications of periodic functions 7.Solutions of simple trigonometric equations within a specified domain 8.Pythagorean identity sin 2 x + cos 2 x = 1

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Radians In the equilateral triangle, each angle is 60 o r r 60 If this chord were pushed onto the circumference, this radius would be pulled back onto the other marked radius 1.Definition of a radian and its relationship to degrees

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Radians 1 radian 57 o 18 2 radians 114 o 36 3 radians 171 o 54 radians = 180 o

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Radians radians = 180 o /2 radians = 90 o /3 radians = 60 o /4 radians = 45 o etc

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Model Express the following in degrees: (a) (b) (c) Remember = 180 o

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Model Express the following in radians: (a) (b) (c) Remember = 180 o

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Exercise NewQ P 294 Set 8.1 Numbers 2 – 5

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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

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3. Definition of the trigonometric functions sin, cos & tan of any angle in degrees and radians 3. Definition of the trigonometric functions sin, cos & tan of any angle in degrees and radians Unit Circle

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Model Find the exact value of: (a) (b) (c)

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45

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Model Find the exact value of: (a) (b) (c) 45

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Model Find the exact value of: (a) (b) (c) 60

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Now lets do the same again, using radians

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Model Find the exact value of: (a) (b) (c)

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Exercise NewQ P 307 Set 9.2 Numbers 1, 2, 8-11

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4. Graphs of y = sin x, y = cos x and y = tan x

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The general shapes of the three major trigonometric graphs y = sin x y = cos x y = tan x

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5. Significance of the constants A,B, C and D on the graphs of… y = A sinB(x + C) + D y = A cosB(x + C) + D

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1.Open the file y = sin(x)Open the file y = sin(x)

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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

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Graph the following curves for 0 x 2 a)y = 3sin(2x) b)y = 2cos(½x) + 1

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Exercise NewQ P 318 Set

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6. Applications of periodic functions

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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

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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

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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 y = a sin b(x+c) + d Tide range = 4m a = 2 Period = 4 Period = 2 /b High tide = 4.5 d = 2.5 b = 0.5

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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 y = 2 sin 0.5(x+c) We need a phase shift of units to the left At the moment, high tide is at hours c =

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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 y = 2 sin 0.5(x+ ) We want the height of the tide when t = 4 On GC, use 2 nd Calc, value h= 1.667m

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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)

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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 sec

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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

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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 Displacement after 10 sec should be the same as displacement after 2 sec = 5.7cm to the left

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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 t = , 11.9, 15.9, , 9.1, 13.1, 17.1

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Exercise NewQ P 179 Set 5.2 1,3

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Model: Find the equation of the curve below. Amplitude = 2.5 y = a sin b(x+c)

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Model: Find the equation of the curve below. Amplitude = 2.5 y = 2.5 sin b(x+c) Period = 6 Period = 2 /b 6 = 2 /b b = /3

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Model: Find the equation of the curve below. Amplitude = 2.5 y = 2.5 sin /3(x+c) Period = 6 Period = 2 /b 6 = 2 /b b = /3 Phase shift = 4 ( ) so c = -4

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Model: Find the equation of the curve below. Amplitude = 2.5 y = 2.5 sin /3(x-4) Period = 6 Period = 2 /b 6 = 2 /b b = /3 Phase shift = 4 ( ) so c = -4

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Exercise NewQ P 183 Set 5.3 1,4

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Find the equation of the curve below in terms of the sin function and the cosine function.

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