2. Errors made by students with Circular Functions 1. Not showing working 2. Not knowing the rules and Exact Values 3. Poor understanding and reasoning ie.Splitting the angle and its coefficient when they should be kept together eg. sin 2θ = 1 then sin θ = 1/2 4. Not checking solutions are in the required domain 5. Errors in calculator use – Use of Radian and Degree Mode NO !!! sin 2θ sin θ
8. Do you know how to use the CAST circle? P() P(-) /2 0 or 2 P(+) P(2-) 3/2
9. Are you familiar with the basic shapes of: y = sinx a = 1 Min is -1 and max is 1. Use this information to draw horizontal lines. Period = 2 Then at max Then at the mean line It starts at the mean line Then at the mean line Then at min
10. What about this? y = cosx a = 1 Min is -1 and max is 1. Use this information to draw horizontal lines. Period = 2 Then at max Then at the mean line It starts at max Then at the mean line Then at min
12. Can you find the asymptotes of tan(bx)? Let Thus Work out the period of tan(bx) Which is Others can be found by adding integer multiples of the period
13. You must know how to solve trigonometric equations in a given domain Draw a CAST circle Tick the two quadrants in which the given function is positive or negative. Find the first quadrant angle, irrespective of the sign. Find the first two solutions between x = 0 and x = 2 (use the appropriate sine, cosine or tangent symmetry property). If more solutions are required: Repeatedly add (or subtract) the period to the two solutions as many times as required, noting solutions after each addition or subtraction. Stop when all solutions within the specified domain are found.
14. You must know the important features of the sine and cosine graphs y = a sin(bx) + c and y = a cos(bx) + c ‘a’ is the dilation factor from the x-axis. The absolute value of ‘a’ gives the amplitude of the graph. ‘1/b’ is the dilation factor from the y-axis. The period of the graph is ‘c’ is the translation factor which moves the graph up or down.
15. The maximum value of the function occurs when sin(bx) and cos(bx) = +1. The minimum value of the function occurs when sin(bx) and cos(bx) = -1. Range = [-a + c, a + c]
16. Problem 1: Heart Rate The heart rate of an athlete during a particular hour of a workout was carefully monitored.
17. Reading from the graph What is the initial heart rate? 110 beats/min. What is the minimum heart rate? 60 beats/min.
18. Finding the rule for this Heart Rate graph This is the amplitude Sine function: H = a sin(bt) + c Determine the values of a, b and c. To determine the amplitude we subtract the minimum point. from the maximum point and divide by 2 . a=(160-60)/2 = 50
19. The mean line is at H = 110 The graph has been translated up 110 c = 110 Period = 60 The period helps us find the ‘b’ value. The graph completes its cycle in 60 seconds. Thus the period is 60. This is the mean line
20. The period is 60 = b = 6 or H = or When modelling with trig functions we generally work with radians unless otherwise specified.
21. Problem 2: Bungee Jumping The height of a bungee jumper, h metres, above a pool of water at any time t seconds after jumping is described by the function: h(t) = 20 cos(0.8t) + 20 What is the initial height of the bungee jumper? When, if at all, does the bungee jumper first touch the water? Assuming the cord is elastic: how long will it be before she returns to the lowest point?
22. Initial Height. The initial height will occur when t = 0 secs Substituting t = 0 into the given equation h(t) = 20 cos(0.8t) + 20 will give us the initial height h(0) = 20 cos(0) + 20 h(0) = 20 x 1 + 20 = 40 metres above the pool of water.
23. Will she hit the pool of water? The minimum of the graph will occur when the cos value is -1. The height of the bungee jumper would then be: 20 x (-1) + 20 = 0. She will hit the water!
24. When will she hit the water? At the minimum point When cos(0.8t) = -1 cos(0.8t) = -1 when 0.8t = t = 3.927 The bungee jumper will first touch the water after 4 seconds (to the nearest second).
25. How long will it be before she returns to the lowest point? From this sketch we can see that she will hit the water again somewhere between 11 and 12 seconds.
26. Solving Trigonometric Equations cos(0.8t) = -1. 0.8t = and 3 Therefore t = 11.79 seconds This will be 8 seconds (to the nearest second) after the first time.
27. Alternatively To find the time that the next minimum occurs we could have added on the period of the graph to 4 seconds. The period is found by Thus the period is 8 seconds. The next time the bungee jumper will touch the water will be 8 seconds later.
28. Applications of sine and cosine graphs Basic graph types are y = a sin(bx) + c and y = a cos(bx) + c To find the maximum value of a function, replace sin(bx) or cos(bx) with +1 To find the minimum value of a function, replace sin(bx) or cos(bx) with -1 Initial values occur at t = 0 A sketch graph may provide greater understanding.
