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

2. 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

3. 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

4. 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

5. 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
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 x
/2 
3/2 2
cos x 1 –1
–4 cos x –4 4 2
y = –2 cos x 4
y = –4 cos x 1
y = –cos x
/2 
3/2 2
/2 
3/2 2
/2 
3/2 2 –1 –¾
y = ¾ cos x –2 –4

6. To be replaced by Ryder 1

7. 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