The document describes the temperature fluctuations in Sumter using a sinusoidal equation. It gives the minimum (-30°C on Dec 31) and maximum (40°C on May 2) temperatures. It then asks to sketch the sinusoidal graph, write the sine and cosine equations identifying the parameters, use one equation to predict the temperature on April 21, find the average temperature, and determine the day(s) when temperature will be 13°C.

1. In Sumter, the coldest temperature drops down to ­30 oC on Dec 31st.
The hottest temperature rises up to 40 oC on May 2nd. Temperature on
other dates can be predicted using a sinusoidal equation.
a) Sketch the graph of the sinusoidal function described above.
b) Write 2 equations for the function; one using sine the other cosine.
c) Use one of the equations in (b) to predict the temperature on April 21st.
d) What is the average temperature throughour the year?
e) On what day will the temperature be at 13 oC?

2. a) Sketch the graph of the sinusoidal function described above.
Amplitude (A)
Vertical Shift (D)
Phase
Shift
(C)
Period = 2 _
_
B

3. b) Write 2 equations for the function; one using sine the other cosine.
Identify the parameters A,B,C,D.
Cosine
Sine
= 35
A = ­35
A
__ __
2π
2π
B= B=
365 365
91
C= C=0
D=
D= 5 5

4. Cosine
Sine
= 35
A = ­35
A
. We found the amplitude (A) by
determining the distance from the
maximum and minimumum values to the
__
average value sinusoidal axis.
. We also know that the graph is stretched by
35 units.

5. Cosine
Sine
= __
2π __
2π
B B=
365 365
We found out what B is by
__
2π 2π
dividing to the period.
B =
Period
The distance along the x­axis
required to make on wave is
Period = __
2π the period
B

6. 2π d
( ) + 5
______
t(d) = ­35cos 365
Sine *Relationship between sine and cosine
= 35
A
__ π
( )
2π
x ­ 2
____
sin(x) = cos
B= 365
π
( )
= 91
C ____
x + 2
cos(x) = sin
D= 5 The graph of sine is equal to the
graph of cosine shifted to the left
π
by .
____
2π
____
2

7. Sine Cosine
D=5
D= 5
D is called the vertical shift a.k.a average value.
It determines the sinusoidal axis.

8. Therefore the equations are:
( ) + 5
2π d
______
t(d) = ­35cos
365
( ) + 5
2π d
______
­91
t(d) = ­35sin
365

9. c) Use one of the equations in (b) to predict the temperature on April 21.
First thing to do is to determine what day is April 21.
Month No. of Days
31
Jan Since where looking
for what day is April
28
Feb 21, we have to
subtract 10 from April
30
Mar because it will be 10
more days if we don't.
31 ­ 10 = 21
Apr
170

10. Now substitute it to one of the equations.
( ) + 5
2π d
______
t(d) = ­35cos
365
t(170) = ­35cos( ) + 5
2π (170)
__________
365
340π
t(170) = ­35cos ( ) + 5
__________
365
t(170) = 39.1928 oC

11. d) What is the average temperature throughour the year?
We already solved for this part. The average
temperature is the same thing as the average value
or the sinusoidal axis which we know is the value
of Parameter D.
Average Temperature = 5 oC

12. e) On what day will the temperature be at 13 oC?
2π d
( ) + 5 8 = ­35cos ( )
2π d
______ ______
t(d) = ­35cos 365 365
13 = ­35cos ( ) + 5
2π d
8 = ­35cosΘ
______
365
13 = ­35cos ( ) + 5
2π d
­ 8 = cos
______
Θ
365 35
8 = ­35cos ( )
2π d
______
Θ= 1.8014, 4.9429
365
let Θ =

13. Θ= 1.8014, 4.9429
let Θ =
Θ= 1.8014 Θ= 4.9429
Θ = Θ =
1.8014 = 4.9429 =
365
______
Multiply both sides by to isolate d.
2π
365 365
______ (1.8014) ______ (4.9429)
d = d =
2π 2π
d = 287
d = 104

14. Therefore the days that experiences a temperature of
13 oC are day 104 and day 287.
April 14 = day 104 October 14 = day 287