Graphs of the Sine and Cosine Functions

Graphs of the Sine and Cosine Functions x x y = sin x 0 0 =SIN(B2) π/6 =PI()/6 =SIN(B3) π/3 =PI()/3 π/2 x x y = sin x 0 0 0.000 π/6 0.524 0.500 =SIN(B4) π/3 1.047 0.866 =PI()/2 =SIN(B5) π/2 1.571 1.000 2π/3 =B5+PI()/6 =SIN(B6) 2π/3 2.094 0.866 5π/6 =B6+PI()/6 =SIN(B7) 5π/6 2.618 0.500 π =B7+PI()/6 =SIN(B8) π 3.142 0.000 7π/6 =B8+PI()/6 =SIN(B9) 7π/6 3.665 -0.500 4π/3 =B9+PI()/6 =SIN(B10) 4π/3 4.189 -0.866 3π/2 =B10+PI()/6 =SIN(B11) 3π/2 4.712 -1.000 5π/3 =B11+PI()/6 =SIN(B12) 5π/3 5.236 -0.866 11π/6 =B12+PI()/6 =SIN(B13) 11π/6 5.76 -0.500 2π =B13+PI()/6 =SIN(B14) 2π 6.283 0.000 If we graph y = sin x by plotting points, we see the following: Going from 0 to 2π, sin(x) starts out with the value 0, then rises to 1 at π/2, then goes back to 0 at π. At x> π, sin(θ) goes from 0 to -1 at 3π/2, then back to 0 at 2π. At this point the sin values repeat. The period of the sine function is 2π. To graph a more complete graph of y= sin x, we repeat this period in each direction. y = sin x 1.500 π/2, 1.000 π/3, 0.866 2π/3, 0.866 1.000 0.500 0.000 -0.500 -1.000 -1.500 π/6, 0.500 0, 0.000 5π/6, 0.500 π, 0.000 2π, 0.000 7π/6, -0.500 11π/6, -0.500 4π/3, -0.866 5π/3, -0.866 3π/2, -1.000 y = sin x 1.500 (π/2, 1) (5π/2, 1) 1.000 0.500 0.000 (0,0) (π, 0) (2π, 0) -0.500 -1.000 (-π/2, -1) (3π/2, -1) -1.500 If we continued the graph in both directions, we’d notice the following: The domain is the set of all real numbers. The range consists of all real numbers such that -1 ≤ sin x ≤ 1. The functions f(x) = sin x is an odd function since the graph is symmetric with respect to the origin. (f(-x) = -f(x) for every x in the domain). The period of the sine function is 2π. The x-intercepts are …., - 2π, -π, 0, π, 2π, 3π, etc.. The y-intercepts is (0,0). The maximum value is 1 and occurs at x = … -3π/2, π/2, 5π/2, etc.. The minimum value is -1 and occurs at x = … -π/2, 3π/2, 7π/2, etc.. Example 1 How do you graph y= sin (x-π/4)? Notice that this function is similar to y= sin x, with (x-π/4) Instead of x. Therefore, this is just a horizontal shift to the RIGHT by π/4 x x y = cos x x 0 0 =COS(B2) π/6 =PI()/6 =COS(B3) π/3 =PI()/3 =COS(B4) π/2 =PI()/2 =COS(B5) 2π/3 =B5+PI()/6 =COS(B6) 5π/6 =B6+PI()/6 =COS(B7) π =B7+PI()/6 =COS(B8) 7π/6 =B8+PI()/6 =COS(B9) 4π/3 =B9+PI()/6 =COS(B10) 3π/2 =B10+PI()/6 =COS(B11) 5π/3 =B11+PI()/6 =COS(B12) 11π/6 =B12+PI()/6 =COS(B13) 2π =B13+PI()/6 =COS(B14) 0 0 π/6 0.524 π/3 1.047 π/2 1.571 2π/3 2.094 5π/6 2.618 π 3.142 7π/6 3.665 4π/3 4.189 3π/2 4.712 5π/3 5.236 11π 5.76 2π 6.283 x y = cos x 1.000 0.866 0.500 0.000 -0.500 -0.866 -1.000 -0.866 -0.500 0.000 0.500 0.866 1.000 If we graph y = cos x by plotting points, we see the following: Going from 0 to 2π, cos(x) starts out with the value 1, then decreases to 0 at π/2, then continues decreasing to -1 at π. At x > π, cos(x) goes from -1 to 0 at 3π/2, then continues increasing to 1 at 2π. At this point the cos values repeat. The period of the cosine function is 2π. To graph a more complete graph of y= cos x, we repeat this period in each direction. y = cos x 1.500 1.000 0.500 0.000 -0.500 -1.000 -1.500 0, 1.000 π/6, 0.866 2π, 1.000 11π/6, 0.866 π/3, 0.500 5π/3, 0.500 π/2, 0.000 3π/2, 0.000 2π/3, -0.500 4π/3, -0.500 5π/6, -0.866 7π/6, -0.866 π, -1.000 y = cos x 1.500 1.000 0, 1.000 -π/6, 0.866 π/6, 0.866 -π/3, 0.500 0.500 -π/2, 0.000 0.000 -2π/3, -0.500 -0.500 -5π/6, -0.866 -1.000 2π, 1.000 11π/6, 0.866 13π/6, 0.866 π/3, 0.500 5π/3, 0.500 π/2, 0.000 3π/2, 0.000 2π/3, -0.500 4π/3, -0.500 7π/3, 0.500 5π/2, 0.000 8π/3, -0.500 5π/6, -0.866 7π/6, -0.866 π, -1.000 -1.500 If we continued the graph in both directions, we’d notice the following: The domain is the set of all real numbers. The range consists of all real numbers such that -1 ≤ cos x ≤ 1. The functions f(x) = cos x is an even function since the graph is symmetric with respect to the y-axis. [f(-x) = f(x) for every x in the domain]. The period of the cosine function is 2π. The x-intercepts are …., - π/2, π/2, 3π/2, 5π/2, etc… The y-intercepts is (0,1). The maximum value is 1 and occurs at x = … -2π, 0, 2π, etc.. The minimum value is -1 and occurs at x = … -π, π, 3π, etc.. 17π/6, -0.866 3π, -1.000 TRANSFORMATIONS y = Asin(ωx + h) + v or y = Acos(ωx + h) + v |A| = amplitude = the biggest value of a periodically changing value. If A < 0, the graph is reflected on the x-axis. -|A| ≤ Asin x ≤ |A| and -|A| ≤ Acos x ≤ |A| [since sin x ≤ 1 and cos x ≤ 1] ω=Frequency = number of periods per time. The number of periods per second is measured in hertz (hz for short). We have a frequency of 1hz if a wave has exactly one period per second. If we have 5 periods per second, we have a frequency of 5hz. Period = the length between two points which are surrounded by the same pattern. For sin and cos functions, period = 2π For tan and cot functions, period = π The new period of this function will either compress or stretch by a factor of 1/ ω. If the 0 < ω < 1, the new period is longer than the original function. If the ω > 1, the new period is shorter than the original function. For sin and cos: T = period of new function = 2π/ω h = horizontal displacement. (+) If h > 0, graph is shifted to the LEFT. (-) If h < 0, graph is shifted to the RIGHT. v = vertical displacement (+) If v > 0, graph is shifted UP. (-) If v < 0, graph is shifted DOWN. y = 5sin (.5x + pi/6)-1 y=sin x 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 π/ 6 π/ π/ 2π 5π π 3 2 /3 /6 3π/ 2 2π 5π/ 2 3π 7π/ 2 4π ⎛ π ⎞ y = 2 sin ⎜ − x ⎟ ⎝ 2 ⎠ Example 8 From the form y = Asin(ωx + h) + v ω = -π/2 (However we need ω > 0) Let’s remember that sin x is an odd function, where f(-x) = -f(x) So ⎛ π ⎞ ⎛π ⎞ y = 2 sin ⎜ − x ⎟ = −2 sin ⎜ x ⎟ ⎝ 2 ⎠ ⎝2 ⎠ Negative sign means sin function will be reflected on the x-axis |A|=2 ω = π/2 The amplitude is |-2| = 2 so the largest value of y is 2. The period is T = 2π/ω = 2π 2 π 2 = 2π • π =4 h = 0, v = 0, so there is no horizontal or vertical displacement. Therefore, the period starts at 0 and ends at 4. sin x = 0 at x = π, so ⎛π sin ⎜ ⎝2 ⎞ x ⎟ = 0, when x = 2 ⎠ y = 2sin (-pi*x/2) 3 2 1 0 -4 -2 0 -1 -2 -3 2 4 6 8 10 12 y = cos x y= sinx 1.500 1.000 0.500 0.000 π/2 π 3π/2 2π 5π/2 3π -0.500 -1.000 -1.500 Notice sin x and cos x are basically the same curves. cos x is just sin x shifted to the left by π/2. Therefore, cos x = sin (x + π/2). Or alternatively, sin x is just cos x shifted to the right by π/2. So sin x = cos (x - π/2). Because of the similarity of cosine and sine curves, these functions are often referred to as sinusoidal graphs.