1. 6-4: Trigonometric
Functions
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Define the Trigonometric
functions in terms of the unit
circle.
•Define the Trigonometric
functions in the coordinate
plane.

2. Important Idea
Trig ratios
depend only
the angle and
not on a point
on the
terminal side
of the angle. θ
(3,4)
(6,8)

3. Example
Findsin ,cosθ θ
& when
the terminal
side of the
angle passes
through (3,4)
tanθ
θ
(3,4)
(6,8)

4. Try This
Findsin ,cosθ θ
& when
the terminal
side of the
angle passes
through (6,8)
tanθ
θ
(3,4)
(6,8)

5. Solution
θ
(6,8)
2 2 2
6 8r = + 10r =
8
6
10
8 4
sin
10 5
θ = =
⇒
6 3
cos
10 5
θ = =
8 4
tan
6 3
θ = =

6. Important Idea
θ
( , )x y
r
x
y
opp
cosθ =
x
r
=
hyp
sinθ =
y
r
=
hyp
adj
tanθ =
opp
adj
y
x
=
See p. 444.
of your text

7. Find sin, cos &
tan of the
angle
whose
terminal side
passes
through the
point (5,-12)
θ
Try This
θ
(5,-12)

8. Solution
θ
5
-12
13
12
sin
13
θ = −
5
cos
13
θ =
12
tan
5
θ = −
(5,-12)

9. Important Idea
Trig ratios may be positive
or negative

10. Find sin, cos &
tan of the
angle
whose
terminal side
passes
through the
point (-5,-5)
θ
Try This
θ
(-5,-5)

11. Solution
θ
(-5,-5)
-5
-5
5 2
5 2
sin
25 2
θ = − = −
5 2
cos
25 2
θ = − = −
5
tan 1
5
θ
−
= =
−

12. Find sin, cos &
tan of the
angle
whose
terminal side
passes
through the
point (5,-12)
θ
Try This
θ
(5,-12)

13. Solution
θ
5
-12
13
12
sin
13
θ = −
5
cos
13
θ =
12
tan
5
θ = −
(5,-12)

14. Example
Find ,sint cost
& when
the terminal
side of an
angle passes
through the
given point on
the unit circle.
tant
1 3
,
10 10
 
− ÷
 
1
10
3
10
−1

15. Important Idea
cos
x
t
r
=
sin
y
t
r
=
tan
y
t
x
=
In the
unit
circle,
r=1,
therefore
1
y
y= =
1
x
x= =
sint y=
cost x=
and

16. Try This
sintFind , cost
when the terminal side of
an angle passes
through
tant&
on the unit circle.
3 4
,
5 5
 
 ÷
 

17. Solution
4
sin
5
t =
3
cos
5
t =
3
35tan
4 4
5
t = =

18. Definition
Coterminal
Angles:
Angles
that have
the same
terminal
side.
x
y
y
x

19. Important Idea
To find coterminal angles,
simply add or subtract
either 360° or 2 radians
to the given angle or any
angle that is already
coterminal to the given
angle.
π

20. Example
Find an
angle
coterminal
with 420°.
Find
sin420°
and
cos420°
1. Find smallest
positive
coterminal angle.
3. Apply
definition of sin
and cos.
Procedure:
2. Draw picture of
coterminal angle.

21. Example
Find an
angle
coterminal
1. Find smallest
positive
coterminal angle.
3. Apply
definition of sin
and cos.
Procedure:
2. Draw picture of
coterminal angle.
7
4
π
−with
Find the
sin and
cos.

22. Important Idea
The trig ratios of a given
angle and all its coterminal
angles are the same.

23. Try This
Find an angle that is
coterminal with 780°. Find
sin780°and cos780°.
3
sin780 sin60
2
° = ° =
1
cos780 cos60
2
° = ° =

24. Try This
Find an angle that is
coterminal with . Find
and .
sin( 10 ) sin0 0π− = =
cos( 10 ) cos0 1π− = =
10π−
sin( 10 )π− cos( 10 )π−
Hint: use the unit circle to
find the trig ratio.

25. Important Idea
In addition to finding trig
ratios of angles ( ), we can
also find trig ratios of real
numbers in radians (t).
Radians may be in terms of
θ
sin
4
π 
 ÷
 
cos( 2.56)− tan
3
π 
 ÷
 
π or just a number, for
example:

26. Important Idea
There are times when we
must be satisfied with
approximate values of trig
ratios. At other times, we
can find and prefer exact
values.

27. Example
cos( 2.56)−
Find the approximate value:
Since the
degree symbol
(°) is not used,
this must be
radians.
mode

28. Try This
Use your calculator
in radian mode to
approximate the
sin, cos and tan.
Round to 4 decimal
places. Use the
signs of the
functions to identify
the quadrant of the
terminal side.
-18
7
8
π
2
5
π
−
35.6π

29. Definition
sint
is the sin of a number
t where t is in radians.
sint =
opposite
hypotenuse
y
r
=
where 2 2
r x y= +
See page 445 of your text.

30. Definition
cost
is the cos of a number
t where t is in radians.
cost =
adjacent
hypotenuse
x
r
=
where
2 2
r x y= +
See page 445 of your text.

31. Definition
tant
is the tan of a number
t where t is in radians.
tant =
opposite
adjacent
y
x
=
See page 445 of your text.

32. Important Idea
cos costθ = =
The definitions of the trig
ratios are the same for
angles and radians, for
example:
sin sintθ = =
hyp
opp y
r
=
hyp
adj x
r
=

33. Example
Find
the
exact
value:
cos45° 45°
cos
4
π 
 ÷
 
10
10
4
π
10
10

34. Example
Find
the
exact
value:
sin30° 1
sin
6
π 
 ÷
 
3
1
6
π
3
30°

35. Definition
Reference Angle: the
angle between a given
angle and the nearest x
axis. (Note: x axis; not y
axis). Reference angles
are always positive.

36. Important Idea
How you find the reference
angle depends on which
quadrant contains the given
angle.
The ability to quickly and
accurately find a reference
angle is going to be
important in future lessons.

37. Example
Find the reference angle if
the given angle is 20°.
In quad. 1,
the given
angle & the
ref. angle are
the same.
x
y
20°

38. Example
Find the reference angle if
the given angle is .
x
y 9
π
9
π In quad. 1,
the given
angle & the
ref. angle are
the same.

39. Example
Find the reference angle if
the given angle is 120°.
For given
angles in quad.
2, the ref. angle
is 180° less the
given angle.
? 120°
x
y

40. Example
Find the reference angle if
the given angle is .
?
x
y
2
3
π
2
3
π
For given
angles in quad.
2, the ref. angle
is less the
given angle.
π

41. Example
Find the reference angle if
the given angle is .
x
y
7
6
π
7
6
π For given
angles in quad.
3, the ref.
angle is the
given angle
less π

42. Try This
Find the reference angle if
the given angle is
7
4
π
For given
angles in quad.
4, the ref. angle
is less the
given angle.
2π
7
4
π
4
π

43. Try This
Find the reference angle if
the given angle is
x
y 4
π
−
Hint: Don’t
forget the
definition.
4
π

44. Important Idea
The trig ratio of a given
angle is the same as the trig
ratio of its reference angle
except, possibly, for the
sign.

45. Example
Find the exact
value of the
sin, cos and tan
of the given
angle in
standard
position. Do
not use a
calculator.
135°

46. Procedure
1.Sketch the given angle.
2.Find and sketch the
reference angle. Label the
sides using special angle
facts.
3.Find sin, cos and tan using
definition.
4.Add the correct sign.

47. Example
Find the exact
value of the
sin, cos and tan
of the given
angle in
standard
position. Do
not use a
calculator.
7
6
π

48. Try This
Find the exact
value of the
sin, cos and tan
of the given
angle in
standard
position. Do
not use a
calculator.
60°

49. 2
Solution
60°
3
1
3
sin60
2
° =
1
cos60
2
° =
tan60 3° =

50. Important Idea
x or y can be positive or
negative depending on
the quadrant but the
hypotenuse ( r ) is
always positive.

51. Try This
Find the exact
value of the
sin, cos and tan
of the given
angle in
standard
position. Do
not use a
calculator.
11
6
π

52. Solution 11
6
π
-1
3
2
11 1
sin
6 2
π 
= − ÷
 
11 3
cos
6 2
π 
= ÷
 
11 1 3
tan
6 33
π 
= − = − ÷
 

53. Try This
Find the exact
value of the
sin, cos and tan
of the given
angle in
standard
position. Do
not use a
calculator.
4
3
π

54. Solution 4
3
π
-1
23−
4 3
sin
3 2
π 
= − ÷
 
4 1
cos
3 2
π 
= − ÷
 
4
tan 3
3
π 
= ÷
 

55. The unit
circle is a
circle with
radius of 1.
We use the
unit circle to
find trig
functions of
quadrantal
angles.
-1 1
-1
1
1
Definition

56. The unit
circle
-1 1
-1
1
1
Definition
(1,0)
(0,1)
(-1,0)
(0,-1)
x y

57. Definition
-1 1
-1
1
(1,0)
(0,1)
(-1,0)
(0,-1)
For the
quadrantal
angles:
The x values
are the terminal
sides for the cos
function.

58. Definition
-1 1
-1
1
(1,0)
(0,1)
(-1,0)
(0,-1)
For the
quadrantal
angles:
The y values
are the terminal
sides for the sin
function.

59. Definition
-1 1
-1
1
(1,0)
(0,1)
(-1,0)
(0,-1)
For the
quadrantal
angles :
The tan function
is the y divided
by the x

60. -1 1
-1
1
Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Example
sinθ
cosθ
tanθ
cscθ
secθ
cotθ
0°
(1,0)
(0,1)
(-1,0)
(0,-1)

61. -1 1
-1
1Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Example
θ
sinθ
cosθ
tanθ
cscθ
secθ
cotθ90°
(1,0)
(0,1)
(-1,0)
(0,-1)

62. -1 1
-1
1
Find the
values of
the six trig
functions of
the given
angle in
standard
position.
2
π
Example
θ
sinθ
cosθ
tanθ
cscθ
secθ
cotθ

63. -1 1
-1
1
Find the
values of
the six trig
functions of
the given
angle in
standard
position.
2π
Example
sinθ
cosθ
tanθ
cscθ
secθ
cotθ

64. -1 1
-1
1
Find the
values of
the six trig
functions of
the given
angle in
standard
position.
3π
Try This
sinθ
cosθ
tanθ
cscθ
secθ
cotθ

65. -1 1
-1
1Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Example
sinθ
cosθ
tanθ
cscθ
secθ
cotθ540°
(1,0)
(0,1)
(-1,0)
(0,-1)

66. -1 1
-1
1Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Example
sinθ
cosθ
tanθ
cscθ
secθ
cotθ270°
(1,0)
(0,1)
(-1,0)
(0,-1)

67. -1 1
-1
1
Find the
values of
the six trig
functions of
the given
angle in
standard
position.
7
2
π
Try This
sinθ
cosθ
tanθ
cscθ
secθ
cotθ

68. -1 1
-1
1Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Try This
sinθ
cosθ
tanθ
cscθ
secθ
cotθ360°
(1,0)
(0,1)
(-1,0)
(0,-1)

69. Important Ideas
•Trig functions of quadrantal
angles have exact values.
•Trig functions of all other
angles have approximate
values.
•Trig functions of special
angles have exact values.

70. Example
Use a calculator to
approximate cos 710° to 4
decimal places.
Don’t forget to check
“Mode”.

71. Example
Use a calculator to
approximate sin(72°30’30”)
to 4 decimal places.

72. Example
Use a calculator to
approximate csc 15° to 4
decimal places.

73. Lesson Close
How do you evaluate the
trig ratios of quadrantal
angles?