+
Starter
Find all values of Θ between 00 and 3600 for which
1. cosΘ = √3/2
Cos Θ is positive in the 1st or 4th quadrant
Θ = 300 , 3300
2. Sin Θ + ½ = 0
Sin Θ = -½
SinΘ is negative in the 3rd and 4th quadrant
Θ = 2100, 3300
+
Note 4: Sine and Cosine Curve
Draw an accurate sketch of the Sine and Cosine
Curve:
y = sin x
y = cos x
for -3600 ≤ Θ ≤ 3600
x-axis - plot every 15°
y-axis from -1 to 1 – plot every 0.1
Sine Curve
Switch your calculator to either
radians or degrees
(Remember 180° =  radians)
Degree
(x)
300
600
900
1200
1500
1800
3600
y = sin x
0.5
0.866
1
0.866
0.5
0
0
y
1
y = sin x
(90, 1)
x
180
–1
360
450
+ Characteristics of the Sine and Cosine Curve

The period is 360° ( how long it takes for the
graph to repeat

The amplitude is 1 ( the height from the
middle

The maximum value is 1 and minimum
value is -1

The domain is: -360° < x < 360°

The range is: -1 < y < 1

The cosine curve is just the sine curve shifted
by 90°
AMPLITUDE, A
y = sin x
y
1
(90, 1)
180
x
360
540
–1
The amplitude is equal to 1.
This is the vertical distance from the equilibrium line (the
x-axis in this case) to a peak or trough
PERIOD T = 360/B (or 2/B if in Raidans)
y = sin x
y
1
0
(/2, 1)

x
2
3
–1
The Period (aka wavelength) is the horizontal distance between
points where the curve repeats itself.
Period equals 2 (or 3600)
This is the length of all orange lines above
+ Investigation 1:
Using technology plot the following:
y = sinx
y = 2sinx
For each graph:

Find the maximum and minimum value

Find the period and amplitude
Describe the effect of A in the function y = Asinx

What is the amplitude of:
y = 4sinx
y = ⅔sinx
y = 2 sin x
2
y
1
0

2
x
–1
y = sin x
–2
2
2
y
2
1
0

2
x
–1
–2
The period of each graph is the same, i.e.
But the amplitude of the new graph is
Y = A sinx
2
2
- Vertical Stretch
+ Investigation 2:
Using technology plot the following:
y = sinx
y = sin2x
For each graph:

Find the maximum and minimum value

Find the period and amplitude
Describe the effect of B in the function y = cosBx

What is the period of:
y = cos4x y = cos¼x
y
y = sin x
1
0
/2

2
–1
y = sin 2x
x
y
y = sin x
1
0
/2

2
–1
x

y = sin 2x
Amplitude is unchanged (1 on both curves)
Y = sinBx
Period is halved, and equals .
- Horizontal Stretch
if |B| > 1 period is shorter
0 < B < 1 period is longer – stretched out
+ Investigation 3:
Using technology plot the following:
y = sinx
y = -2sinx, y = 2 sinx
y = sin(-2x), y = sin2x
Describe the effect of the negative in the trig
functions.
The negative means a reflection in the x-axis
+ Investigation 4:
y = sin(x) + 2
y = sin(x) – 1

Calculate the equation of the principal axis
y = sin(x) + D
Vertical Translation
if D is positive – shift up
D is negative – shift down
Equation of principal axis (new x axis):
y=D
The original curve is moved vertically up 2 units.
Amplitude & period unchanged
y
4
y = sin x + 2
3.5
3
2.5
2
y = sin x
1.5
1
(“original”)
0.5
x
1
-0.5
-1
2
3
4
5
6
7
8
9
10
+ Investigation 5:
y = sin(x - 450)
y = sin(x + 600)
y = sin(x - C)
Horizontal Translation
if C is positive – shift curve to right
if C is negative – shift curve to left
+ IN GENERAL:
y = AsinBx + D
Affects
Amplitude
Affects
Period
Affects Principal Axis
i.e Vertical Translation
y = AsinB(x – C) + D
Affects horizontal
translation
To find:


Period = 360/B for degrees, 2π/B for radians
Principal axis y = D
Complete the table
Graph
Amplitude, A
Period, T
y= sin x
1
2
y= cos x
y= 5sin x
1
2
5
2
3 & is upside
down
y= – 3 cos x
2
y= sin 7x
1
y= cos ¼x
1
2/ ¼ i.e. 8
y= 3sin 2x
3
2/2 i.e. 
y= 2 cos
x
3
2
2/7
2/1 i.e. 6
3
Example. Find the period and amplitude
of y = 3sin x
Write down: a = 3 and b = 
Amplitude (A) is equal to 3.
Period (T) is equal to 2/B i.e. 2/ = 2
Example. Find the period and
amplitude of y = – 5 cos (x/6)
Write down: a = 5 and B = /6
Noting that
x
6
Amplitude (A) is equal to 5.
Period (T) is equal to 2/(/6) i.e. 12


6
x
Example. Find A , T and B and hence the equation of these
graphs:
y
2
x
4
1. Goes through 0, so its a sine
2. Amplitude = 2. i.e. A = 2
3. Period T = 4 .
4. T = 2 /B so 4 = 2 /B
B = 2 /4 
B = 0.5.
y = 2 sin 0.5x
Page 273
Exercises 13D.1
13D.2