### PC 01-26n27 Translations of Sine and Cosine Functions

```Trigonometry, 4.0: Students graph functions of the form
f(t)=Asin(Bt+C) or f(t)=Acos(Bt+C) and interpret A, B, and C
in terms of amplitude, frequency, period, and phase shift.
1.
State the amplitude and period for each function. Then
graph each function.
a) y=-3 cos(2θ)
b) y=2/3 cos(θ/4)
c) y=sin(4θ)
Write an equation of the sine function with amplitude
0.27 and period π/2.
3. Write an equation of the sine function with amplitude
3/5 and period 4.
3, π 2/3, 8π
1, π/2
y=±0.27 sin(4θ)
y=±3/5 cos(π/2 θ)
2.
Find the phase shift translations for sine and cosine
functions.
2. Find the vertical translations for sine and cosine
functions.
3. Write the equations of sine an cosine functions given
the amplitude, period, phase shift, and vertical
translation
4. Graph compound functions.
1.
Phase shift of Sine and Cosine Functions:
y=A sin[B(θ-h)]+k and y=A cos[B(θ-h)]+k
 The horizontal shift is h
 If h>0, the shift is to the right
 If h<0, the shift is to the left
State the phase shift for each function. Then graph
the function.
 a. y = sin (2 + )
 b. y = cos ( - )
 A. y = sin (2 + )
The phase shift of the function is
−

.
2

−

or
To graph y = sin (2 + ), consider the
graph of y = sin 2. Graph this function

and then shift the graph − .
2
 B. y = cos ( - )
The phase shift of the function is
−

,
1
which equals .

−

To graph y = cos ( - ), consider the
graph of y = cos  and then shift the
graph .
or
Vertical shift of Sine and Cosine Functions:
y=A sin[B(θ-h)]+k and y=A cos[B(θ-h)]+k
 The midline is y=k
 If k>0, the shift is upward
 If k<0, the shift is downward
State the vertical shift and the equation of the
midline for the function y = 3 cos  + 4.
 Then graph the function.
 The vertical shift is 4 units upward. The
midline is the graph y = 4.
 To graph the function, draw the midline,
the graph of y = 4. Since the amplitude
of the function
 is 3, draw dashed lines parallel to the
midline which are 3 units above and
below the midline. Then draw the cosine
curve.
Graphing Sine and Cosine Functions:
1. Determine the vertical shift and graph the midline.
2. Determine the amplitude. Use dashed lines to
indicate the maximum and minimum values of the
function.
3. Determine the period of the function and graph the
appropriate sine or cosine curve.
4. Determine the phase shift and translate the graph
accordingly.
State the amplitude, period, phase shift, and
vertical shift for y = 2 cos ( /2 + ) + 3.
 The amplitude is 2 or 2. The period is
 The phase shift is −

1
2
2
1
2
or 4.
or -2. The vertical shift is +3
Write an equation of a sine function with amplitude 5, period
3, phase shift /2, and vertical shift 2.
 The form of the equation will be y = A sin (k + c) + h. Find the
values of A, k, c, and h.
 A:
|A|
=5
A
= 5 or -5
 k:
2/k = 3 The period is 3.
k
= 2/3
 c:
-c/k
= /2 The phase shift is /2.
-c/ 2/3 = /2 k = 2/3
c
= - /3
 h:
h=2
 Substitute these values into the general equation. The possible
equations: y = 5 sin (2/3 - /3) + 2 or y = - 5 sin (2/3 - /3) + 2
 Compound functions may consist of sums or products
of trigonometric functions or other functions.
 For example:
  = sin  ∙ cos  Product of trigonometric functions
  = cos  +
linear function.
Sum of a trigonometric function and a
 Graph y = x + sin x.
 First graph y = x and y = sin x on the same axes. Then
add the corresponding ordinates of the functions.
Finally, sketch the graph.
x
sin x x + sin x
0
0
0
1
+ 1 2.57
/2
0
 3.14

-1
- 1 3.71
3/2
0
2
2 6.28
1
+ 1 8.85
5/2
0
3
3 9.42
Summary
Assignment
 A Japanese company invented
 6.5 Translations of Sine
in 1966. Suppose that
researchers were observing a
sine curve that had an amplitude
of 3 centimeters, a period of 9
centimeters, an upper shift of 2
centimeters, and a phase shift ½
centimeter to the right. State the
function that models the data.
 y=3 sin(2π/9 θ - π/9) + 2, this
is one of many equivalent