7.1: Graphs of the Sine,
Cosine & Tangent
Functions
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Graph Trig func...
Example
Using a
table of
values,
graph
siny t=
for
2 2tπ π− ≤ ≤
2π−
3 2π−
π−
2π−
0
2π
π
3 2π
2π
Try This
Using a
table of
values,
graph
cosy t=
for
2 2tπ π− ≤ ≤
2π−
3 2π−
π−
2π−
0
2π
π
3 2π
2π
2π−
Solution
2π0
t
Example
Using your graphing
calculator, graph ( ) cosg t t=
on the interval [ ]3 ,2π π−
Try ThisOn a
sheet of
paper,
compare
and
contrast
the
graphs:siny t=
cosy t=
Example
cosy t=
1
-1
0
Animate
Example
sint
Without using your graphing
calculator, find all values of
t for which is 1.
Try This
cost
Without
using your
graphing
calculator,
find all
values of t
for which
is 1.
0 2t kπ= +
k=integer
ExampleWithout
using your
calculator,
find all
values of t
between 0
and 2 for
which
π
sint
= 1
2
2π0
1
2
Try This
Without using your
calculator, find all values
of t between 0 and 2 for
which
π
cost= 1
2
5
,
3 3
t
π π
=
Analysis
( ) cosf t t=For :
•What is the domain and
range?
•What is the unit of measure
for the t and axes?( )f t
Definition
The Domain of a function is
all possible values of the
independent variable (t,x,θ)
The Range of a function is
...
Analysis
( ) cosf t t=( ) sinf t t=
Compare and contrast
the two graphs.
+1
-1
Try This
What is
wrong with
these
statements: 3
cos
2
t = −
sin 2t =
Try This
Graph the function :
( ) tanf x x=
using your graphing
calculator
Solution
( ) tanf x x=
What is the
domain and
range?
What are
Asymptotes
and where
do they
occur?
Important Idea
Asymptotes
occur at all
points in the
domain
where the
function is
not defined.
(0,-1)
(-1,0) (1,0)
(0,1)
Important Idea
Asymptotes
occur at all
points in the
domain
where the
function is
not defined.
(0,-1)
(-1,0) (1,0)
(0,1)
Try ThisCompare
and
contrast
the
graphs:
siny t= −
siny t=
1
-1
Try ThisCompare
and
contrast
the
graphs:
siny t=
sin 1.5y t= +
2.5
1
Example
For what
values of t on
the interval
[ ]2 ,2π π− is:
( )f t• increasing
• decreasing( )f t
• ( ) 0f t < 2π− 2π
( )...
Example
0 cos 1t< <
For what
values of t on
the interval
[ ]3 ,2π π− is:
3π− 2π
( ) cosf t t=
Example
3
cos
2
t = −
For what
values of t on
the interval
[ ]2 ,2π π− is:
2π− 2π
( ) cosf t t=
3
2
−
Example
3
cos
2
t = −
Name all,
exact
values of t
for which
( ) cosf t t=
3
2
−
Try This
1
sin
2
t =
Name all,
exact
values of t
for which
1
2
( ) sinf t t=
2
6
t n
π
π= +
5
2
6
t n
π
π= +
or
Lesson Close
Define:
•Domain
•Range
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Hprec7.1

  1. 1. 7.1: Graphs of the Sine, Cosine & Tangent Functions © 2008 Roy L. Gover(www.mrgover.com) Learning Goals: •Graph Trig functions •Analyze and interpret Trig graphs
  2. 2. Example Using a table of values, graph siny t= for 2 2tπ π− ≤ ≤ 2π− 3 2π− π− 2π− 0 2π π 3 2π 2π
  3. 3. Try This Using a table of values, graph cosy t= for 2 2tπ π− ≤ ≤ 2π− 3 2π− π− 2π− 0 2π π 3 2π 2π
  4. 4. 2π− Solution 2π0 t
  5. 5. Example Using your graphing calculator, graph ( ) cosg t t= on the interval [ ]3 ,2π π−
  6. 6. Try ThisOn a sheet of paper, compare and contrast the graphs:siny t= cosy t=
  7. 7. Example cosy t= 1 -1 0 Animate
  8. 8. Example sint Without using your graphing calculator, find all values of t for which is 1.
  9. 9. Try This cost Without using your graphing calculator, find all values of t for which is 1. 0 2t kπ= + k=integer
  10. 10. ExampleWithout using your calculator, find all values of t between 0 and 2 for which π sint = 1 2 2π0 1 2
  11. 11. Try This Without using your calculator, find all values of t between 0 and 2 for which π cost= 1 2 5 , 3 3 t π π =
  12. 12. Analysis ( ) cosf t t=For : •What is the domain and range? •What is the unit of measure for the t and axes?( )f t
  13. 13. Definition The Domain of a function is all possible values of the independent variable (t,x,θ) The Range of a function is all possible values of the dependent variable , ( ), ( )y f t f θ
  14. 14. Analysis ( ) cosf t t=( ) sinf t t= Compare and contrast the two graphs. +1 -1
  15. 15. Try This What is wrong with these statements: 3 cos 2 t = − sin 2t =
  16. 16. Try This Graph the function : ( ) tanf x x= using your graphing calculator
  17. 17. Solution ( ) tanf x x= What is the domain and range? What are Asymptotes and where do they occur?
  18. 18. Important Idea Asymptotes occur at all points in the domain where the function is not defined. (0,-1) (-1,0) (1,0) (0,1)
  19. 19. Important Idea Asymptotes occur at all points in the domain where the function is not defined. (0,-1) (-1,0) (1,0) (0,1)
  20. 20. Try ThisCompare and contrast the graphs: siny t= − siny t= 1 -1
  21. 21. Try ThisCompare and contrast the graphs: siny t= sin 1.5y t= + 2.5 1
  22. 22. Example For what values of t on the interval [ ]2 ,2π π− is: ( )f t• increasing • decreasing( )f t • ( ) 0f t < 2π− 2π ( ) cosf t t=
  23. 23. Example 0 cos 1t< < For what values of t on the interval [ ]3 ,2π π− is: 3π− 2π ( ) cosf t t=
  24. 24. Example 3 cos 2 t = − For what values of t on the interval [ ]2 ,2π π− is: 2π− 2π ( ) cosf t t= 3 2 −
  25. 25. Example 3 cos 2 t = − Name all, exact values of t for which ( ) cosf t t= 3 2 −
  26. 26. Try This 1 sin 2 t = Name all, exact values of t for which 1 2 ( ) sinf t t= 2 6 t n π π= + 5 2 6 t n π π= + or
  27. 27. Lesson Close Define: •Domain •Range

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