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Basic Graphs of Sine and Cosine Functions 4.1 JMerrill, 2009 (contributions by DDillon) Sine Function x 0 y 0 6 4 3 2 2 3 1 2 2 2 3 2 1 3 2 3 5 4 6 7 5 6 4 4 3 5 7 11 3 2 3 4 6 2 1 2 0 1 2 3 -1 3 2 1 2 2 2 2 2 2 0 Notice the sine function has origin symmetry. (If you rotate it 180° about the origin, the graph looks the same.) This means that the sine function is odd. sin (-x) = - sin x 2 2 Period: Sine Function x 0 6 4 3 2 2 3 y 1 2 2 2 3 2 1 3 2 0 This one piece of the sine function repeats over and over, causing the sine function to be periodic. The length of this piece is called the period of the function. 3 5 4 6 7 5 6 4 4 3 5 7 11 3 2 3 4 6 2 1 2 0 1 2 3 -1 3 2 1 2 2 2 2 2 2 0 2 2 Cosine Function x 0 y 1 6 4 3 3 2 2 2 1 2 2 2 3 0 1 2 3 2 2 2 Notice the cosine function has y-axis symmetry. (If you reflect it across the yaxis, the graph looks the same.) This means that the cosine function is even. cos (-x) = cos x 3 5 4 6 7 5 6 4 4 3 5 7 11 3 2 3 4 6 -1 3 2 1 2 2 2 0 1 2 2 2 3 2 2 1 Period: Cosine Function x 0 y 1 6 4 3 3 2 2 2 1 2 2 2 3 3 5 4 6 0 1 2 3 2 2 2 This one piece of the cosine function repeats over and over, causing the cosine function to be periodic. The length of this piece is called the period of the function. 7 5 6 4 4 3 5 7 11 3 2 3 4 6 -1 3 2 1 2 2 2 0 1 2 2 2 3 2 2 1 Period The period of a normal sine or cosine function is 2π. To change the period of a sine or cosine function, you would need to horizontally stretch or shrink the function. The period is found by: period = y sin Bx 2 B y cos Bx Period Examples of f(x) = sin Bx • The period of the sin(x) (parent) is 2π • The period of sin2x is π. p= 2 2 b 2 • If B > 1, the graph shrinks. • This graph is happening twice as often as the original wave. Period Examples of f(x) = sin Bx • The period of the sinx (parent) is 2π • The period of sin ½ x is 4π. p= 2 1 2 2 2 4 • If b < 1, the graph stretches. • This graph is happening half as often as the original wave. What is the period? Examples y sin 1 x 2 y cos 3x 2 y sin x 3 y cos 2 x Horiz. stretch by ½ period 2 2 4 Horiz. shrink by 3 2 period 2 3 3 Horiz. shrink by 2π/3 2 period 2 3 3 Horiz. shrink by π/2 period 2 2 4 Amplitude: Sine Function x 0 6 4 3 2 2 3 y 1 2 2 2 3 2 1 3 2 0 The maximum height of the sine function is 1. It goes one unit above and one unit below the x-axis, which is the center of it’s graph. This maximum height is called the amplitude. 3 5 4 6 7 5 6 4 4 3 5 7 11 3 2 3 4 6 2 1 2 0 1 2 3 -1 3 2 1 2 2 2 2 2 2 0 2 2 1 1 Amplitude: Cosine Function x 0 y 1 6 4 3 3 2 2 2 1 2 2 2 3 0 1 2 3 2 2 2 The maximum height of the cosine function is 1. It goes one unit above and one unit below the xaxis, which is the center of it’s graph. This maximum height is called the amplitude. 3 5 4 6 7 5 6 4 4 3 5 7 11 3 2 3 4 6 -1 3 2 1 2 2 2 0 1 2 2 2 3 2 1 1 2 1 Amplitude The amplitude of the normal sine or cosine function is 1. To change the amplitude of a sine or cosine function, you would need to vertically stretch or shrink the function. amplitude = |A| (Choose the line that is dead-center of the graph. The amplitude has the same height above the center line (axis of the wave) as the height below the center line. y A cos x y A sin x What is the amplitude? Examples y 3sin x y y 1 cos x 2 4 sin x Vert. stretch by 3 Vert. shrink by ½ Vert. shrink by π/4 amp. 3 1 amp. 2 amp. 4 Examples: y = A sin Bx y = A cos Bx • Give the amplitude and period of each funtion: Y = 4 cos 2x A = 4, p 2 2 y= -4 sin 1/3 x A = 4, p 2 2 (3) 6 1 3 3 sin x 2 2 3 2 2 A ,p 2 4 2 2 y Can You Write the Equation? • • • • Sine or cosine? Amplitude? 2 Period? 24 b? 1 12 • Equation? 1 y 2cos x 12 Equation? • • • • Sine or Cosine? Amplitude? 2 Period? 8 b? 4 • Equation: y 2 sin 4 x Harmonic Motion • • • • 3 Types: Simple – unvarying period motion Damped – motion decreases with time Resonance – motion increases with time Weight on Spring video A weight is at rest hanging from a spring. It is then pulled down 6 cm and released. The weight oscillates up and down, completing one cycle every 3 seconds. Sketch Distance above/below resting point, in cm 6 3 -6 Time, in seconds Equation Amplitude = 6 6 A=6 3 = 2π/B B = 2π/3 3 -6 2 y 6sin ( x 1.5) 3 Positions Determine the position of the weight at 1.5 seconds. Let x = 1.5; plug into equation for function. y = 0 cm (back at original position) Use the graph to find the time when y = 3.5 for the first time. Graph y1 = equation you wrote; graph y2 = 3.5. Find intersection. x = 1.797 seconds 3.5 is the 3.5 cm distance above the original position of the weight.