Periodic Functions & Applications II Topic 4 Periodic Functions & Applications II 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

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

Radians 1 radian 57o18’ 2 radians 114o36’ 3 radians 171o54’

Radians  radians = 180o /2 radians = 90o /3 radians = 60o etc

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

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

Exercise NewQ P 294 Set 8.1 Numbers 2 – 5

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) (b) (c)

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

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

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

Now let’s do the same again, using radians

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

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

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

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

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, C and D on the graphs of… y = A sinB(x + C) + D y = A cosB(x + C) + D

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 y = 3sin(2x) 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 Find the height of the tide at 4am y = a sin b(x+c) + d Tide range = 4m  a = 2 High tide = 4.5  d = 2.5 Period = 4 Period = 2/b  b = 0.5

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) + 2.5 At the moment, high tide is at  hours We need a phase shift of  units to the left  c = 

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+) + 2.5 We want the height of the tide when t = 4 On GC, use 2nd Calc, value  h= 1.667m

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)

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 = 4 sec

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

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

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 = 1.1 5.1, 9.1, 13.1, 17.1 3.9 7.9, 11.9, 15.9, 19.9

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

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

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

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

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.