Sketching y = a sin bx and y = a cos bx

y = sin x y = cos x x 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 sin x .5 .71 .87 1 –.5 –.71 –.87 –1 90 180 270 360 1 –1 0.5 –0.5 45 135 225 315 y = cos x x 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 cos x 1 .87 .71 .5 –.5 –.71 –.87 –1 90 180 270 360 1 –1 0.5 –0.5 45 135 225 315

y = sin x y = cos x x –2 –3/2 – –/2 /2  3/2 2 5/2 3 7/2 4 /2  3/2 2 5/2 3 7/2 4 sin x 1 –1 1 y = sin x and y = cos x are called periodic functions because of the repeating cycles. For sine, it’s For cosine, it’s: – –3/2 – –/2 /2  3/2 2 5/2 3 7/2 4 –1 y = cos x x –2 –3/2 – –/2 /2  3/2 2 5/2 3 7/2 4 cos x 1 –1 1 –2 –3/2 – –/2 /2  3/2 2 5/2 3 7/2 4 –1

y = –sin x  Reflect y = sin x over the x-axis /2  3/2 2 sin x 1 –1 x /2  3/2 2 –sin x –1 1 1 1 /2  3/2 2 /2  3/2 2 –1 –1 y = cos x y = –cos x  Reflect y = cos x over the x-axis x /2  3/2 2 cos x 1 –1 x /2  3/2 2 –cos x –1 1 1 1 /2  3/2 2 /2  3/2 2 –1 –1

Sketching y = a sin x y = 2 sin x y = ½ sin x y = –3 sin x /2  3/2 2 sin x 1 –1 2 sin x 2 –2 x /2  3/2 2 sin x 1 –1 ½ sin x 1/2 –1/2 x /2  3/2 2 sin x 1 –1 –3sinx –3 3 2 1 3 y = 2 sin x y = ½ sin x 1 ½ y = sin x /2  3/2 2 /2  3/2 2 /2  3/2 2 –1 –½ –2 –1 –3 Sketching y = a cos x y = –2 cos x y = ¾ cos x y = –4 cos x x /2  3/2 2 cos x 1 –1 –2 cos x –2 2 x /2  3/2 2 cos x 1 –1 ¾ cos x ¾ –¾ x /2  3/2 2 cos x 1 –1 –4 cos x –4 4 2 y = –2 cos x 4 y = –4 cos x ¾ 1 y = –cos x /2  3/2 2 /2  3/2 2 /2  3/2 2 –1 –¾ y = ¾ cos x –2 –4

To be replaced by Ryder 1

Sketching y = a sin bx and y = a cos bx: |a| = amplitude = ½(ymax – ymin) i.e., the “height” from the middle to the top |b| = frequency = number of cycles in 2 units P = period = the least number of positive units it takes to complete one cycle = 2/|b| Plot the two functions f(x) and g(x) on the same graph on 0  x  2. For each function, identify the amplitude, frequency, period and answer the number of times they intersect. f(x) = 2 cos (3x) Amplitude: 2 Frequency: 3 Period: 2/3 g(x) = –2 sin (¾x) Amplitude: 2 Frequency: 3/4 Period: 2/(¾) = 8/3 2 1 2 3 –1 –2 –3 g(x) = –2 sin (¾x) f(x) = 2 cos (3x) How many times do they intersect on the interval [0, 2]? 6 times Give a solution for the equation 2 cos (3x) = –2 sin (¾x) on [0, 2]: 2