Graphs of Sine and Cosine
Functions
Lesson 2.5
Ordered Pairs


Consider the values
for x and y in the
table to the right
Note



Period = 2π
Maximum y values
Minimum y values
x
sin(x)
cos(x)
-3.1416
0.0000
-1.0000
-2.6180
-0.5000
-0.8660
-2.0944
-0.8660
-0.5000
-1.5708
-1.0000
0.0000
-1.0472
-0.8660
0.5000
-0.5236
-0.5000
0.8660
0.0000
0.0000
1.0000
0.5236
0.5000
0.8660
1.0472
0.8660
0.5000
1.5708
1.0000
0.0000
2.0944
0.8660
-0.5000
2.6180
0.5000
-0.8660
3.1416
0.0000
-1.0000
3.6652
-0.5000
-0.8660
4.1888
-0.8660
-0.5000
4.7124
-1.0000
0.0000
5.2360
-0.8660
0.5000
5.7596
-0.5000
0.8660
6.2832
0.0000
1.0000
2
Graphing the Ordered Pairs
1.5
Period = 2π
1.0
Maximum and
minimum values
-6.28
-3.14
0.5
0.0
0.00
-0.5
sin(x)
3.14
6.28
9.42
cos(x)
-1.0
-1.5
3
Graphing on Calculator

Go to ♦Y= screen

Enter function

Choose F2,
zoom 7-Trig

Graph is plotted

Tic marks are in
units of π/2
Try Web Graphing
Utility
4
Amplitude


Defined as the absolute
value of maximum
or minimum of the
function
amplitude = 1
Try graphing
y = 2 cos x


What is the amplitude
For y = a cos x or y = a sin x

The amplitude is |a|
5
Period of a Trig Function
(Recall slide from previous lesson)



The functions repeat themselves
The period is the smallest value, p for
which
f(x) = f(x + p)
For sin, cos, sec, csc


The period is 2π
For tan and ctn

The period is π
6
Period of a Trig Function


What happens for y  sin  b  x  ?
Try graphing y = sin 3x



What is the period?
Try y = cos 0.5x

What is the period?

2
Period =
b
For y  sin  b  x 
Same for cos, sec, csc
7
Period of a Trig Function

For tangent



Note amplitude
is without bound
Period is π
y  tan x
For y  tan  b  x 

Period =

b
• Predict the period for
y = tan (1/3 x)
• Graph it and verify your
prediction
8
Assignment



Lesson 2.5
Page 177
Exercises 1 – 61 EOO
also 63
9