5.1: Graphing Sine and Cosine Functions

Transcription

1 5.1: Graphing Sine and Cosine Functions Complete the table below ( we used increments of for the values of ) 4 0 sin cos 1. Using the table, sketch the graph of y sin for What happens as continues beyond 2? The Graph of the Function y sin, for Using the table, sketch the graph of y cos for What happens as continues beyond 2? The Graph of the Function y cos ( for 0 2 ) What is the maximum value of What is the minimum value of What are the domain and range of sin and sin and cos? cos? sin and cos? 1

2 Period : Trigonometric functions are periodic functions. The graph of a periodic function repeats in a regular way. The length of the shortest part that repeats, measured along the horizontal axis, is called the period of the function. What is the period of Amplitude: m M sin and a cos? The number a is the amplitude of a trigonometric function. It represents the distance from any maximum or minimum to the mid-line. M m a 2 where M = maximum value m = minimum value Ex 1. a) Graph the functions y sin x and y cos x on the same axis for 2 x 2. b) How do the two functions appear to be related? Explain. c) What are the coordinates of the points of intersection? (use exact values) Ex 2. a) Graph the functions y sin x, y 3sin x and y 0.5sin x on the same axis from 0 x 2. (Check with a graphing calculator) b) State the amplitude of each function. 2

3 For the functions of the forms y asin b and y acosb, where a, b 0, the amplitude is a, and the period is or b b Ex 3. Give the amplitude and period : Ex 4. Determine the amplitude and the period in radians of the function function to verify your answers. x y 2cos. Graph this 3 Ex 5. Graph one complete cycle 1 a) y sin 2x b) y cos3x 2 Ex 6. Write an equation of the a) sine fct with amplitude 2.8 and period 60 b) cosine fct with amplitude 3, and period 2 3

4 Ex 7. What is a possible equation of each of the following functions? a) b) 5.2: Transformations of Sinusoidal Functions Sinusoidal functions are functions that oscillate up and down as a sine or cosine function. We can transform sinusoidal functions just as we transformed many other functions. Ex. 1: a) Sketch the graph of the function y sin(x 30)2 b) What are the domain and the range of the function? Vertical displacement: the vertical translation of the graph of a periodic function This will be the center line of your function. Phase shift: the horizontal translation of the graph of a periodic function This will be the starting point of your function. 4

5 Ex. 2: a) Sketch the graph of the function y 2cos( )1 over two cycles. b) Use the language of transformations to compare your graph to the graph of y cos. Graphing Sinusoidal Functions: 1. Use the Vertical Displacement and Phase Shift to find your Starting Point. 2. Determine the Amplitude to find your Max/Min Values. 3. Determine the period and place your Max/Min on your graph. Sketch. Ex. 3: Sketch the graph of the function y 3sin2(x )2 over two cycles. 3 vertical displacement: phase shift : amplitude: period: maximum: minimum: 5

6 Ex. 4: The graphs shows the function y f (x). a) Write the equation of the function in the form b) Write the equation of the function in the form y asinb(x c) d y acosb(x c) d Summary: y asinb(x c) d y acosb(x c) d a = amplitude (vertical stretch) b = reciprocal of period (horizontal stretch) where p = 2 b c = phase shift starting point d = vertical displacement centre line 5.3: The Tangent Function Draw a diagram of a point on the unit circle in quadrant 1 below to discuss the tangent ratio in terms of sine and cosine. 6

7 Ex. 1: Graph the function y tanx for 2 x 2. Describe its characteristics. amplitude = period = asymptotes = domain = range = Ex. 2: Graph the function y tan(x )1 over two cycles. Describe its characteristics. 2 amplitude = period = asymptotes = domain = range = 7

8 Ex. 3: A small plane is flying at a constant altitude of 6000 m directly toward an observer. Assume that the ground is flat in the region close to the observer. a) Determine the relation between the horizontal distance, in metres, from the observer to the plan and the angle, in degrees, formed from the vertical to the plane. b) Sketch the graph of the function. c) Where are the asymptotes located in this graph? What do they represent? d) Explain what happens when the angle is equal to 0. 8

9 Applications of Sinusoidal Functions Ex 1: A Ferris wheel has a radius of 42 m. Its centre is 43 m above the ground. It rotates once every 50 s. Suppose you get on at t 0. a) Graph how your height above the ground varies during the first two cycles. b) Write an equation that expresses your height as a function of the elapsed time, in the form h( t) Asin( B( t C)) D c) Estimate your height above the ground after 65 s. d) Estimate one of the times when your height is 25 m above the ground. Ex 2: At a seaport, the water has a maximum depth of 15 m at 7:00 a.m. The minimum depth of 5 m occurs 6.2 hours later. Assume the relation between the depth of the water and time is a sinusoidal function. a) What is the period of the function? b) Write an equation for the depth, h meters, of the water at any time, t hours. c) Estimate the depth at 11:00 a.m. d) Estimate one of the times when the water is 11 m deep. 9

10 Ex 3: The height, h meters, of a person in a Ferris wheel is a sinusoidal function of time t seconds. A graph illustrating variations in height is shown. Write an equation for this function. Ex. 4: Ferris wheel has a radius of 18 metres and a centre C which is 20 m above the ground. It rotates once every 32 seconds in the direction shown in the diagram. A platform allows a passenger to get on the Ferris wheel at a point P which is 20 m above the ground. If the ride begins at point P, when the time t = 0 seconds, determine a sine function that gives the passenger s height, h, in metres, above the ground as a function of t. A. h ( t) 18sin t B. h ( t) 18sin t C. h ( t) 20sin t D. h ( t) 20sin t

11 5.4: Equations and Graphs of Trigonometric Functions Trigonometric functions are often used to model different phenomena that have wave characteristics. We can use the graphs to help us find solutions to a corresponding equation by graphing and finding any zeros. Observe this example of pendulums! The pendulums are all periodic and repeat their patterns over a certain period. The lengths of the ropes they hang from are all different, altering the period slightly. (Cool!) Ex. 1: Determine the solutions for the trigonometric equation 0 x cos 2 x 1 0 for the interval a) Graphical solution b) Algebraic solution Ex. 2: Determine the general solutions for the trigonometric equation 16 6cos x 14. Express your 6 answers to the nearest hundredth. Method 1: Find Zeros Method 2: Find Intersections Y 1 = Y 1 = Y 2 = x [, ] y[, ] x [, ] y[, ] 11

12 Method 3: Algebraically Ex. 3: The electricity coming from power plants into your house is alternating current (AC). This means that the direction of current flowing in a circuit is constantly switching back and forth. In Canada, the current makes 60 complete cycles each second. The voltage can be modelled as a function of time using the sine function V 170sin120t a) What is the period of the current in Canada? b) Graph the voltage function over two cycles. Explain what the scales on the axes represent. Y 1 = x [, ] y[, ] c) Suppose you want to switch on a heat lamp for an outdoor patio. If the heat lamp requires 110 V to start up, determine the time required for the voltage to first reach 110 V. 12