6-4: Trigonometric
Functions
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Define the Trigonometric
functions in ...
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)
Example
Findsin ,cosθ θ
& when
the terminal
side of the
angle passes
through (3,4)
tanθ
θ
(3,4)
(6,8)
Try This
Findsin ,cosθ θ
& when
the terminal
side of the
angle passes
through (6,8)
tanθ
θ
(3,4)
(6,8)
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 = =
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
Find sin, cos &
tan of the
angle
whose
terminal side
passes
through the
point (5,-12)
θ
Try This
θ
(5,-12)
Solution
θ
5
-12
13
12
sin
13
θ = −
5
cos
13
θ =
12
tan
5
θ = −
(5,-12)
Important Idea
Trig ratios may be positive
or negative
Find sin, cos &
tan of the
angle
whose
terminal side
passes
through the
point (-5,-5)
θ
Try This
θ
(-5,-5)
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
−
= =
−
Find sin, cos &
tan of the
angle
whose
terminal side
passes
through the
point (5,-12)
θ
Try This
θ
(5,-12)
Solution
θ
5
-12
13
12
sin
13
θ = −
5
cos
13
θ =
12
tan
5
θ = −
(5,-12)
Example
Find ,sint cost
& when
the terminal
side of an
angle passes
through the
given point on
the unit circle.
tant
1 3
,...
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
Try This
sintFind , cost
when the terminal side of
an angle passes
through
tant&
on the unit circle.
3 4
,
5 5
 
 ÷
 
Solution
4
sin
5
t =
3
cos
5
t =
3
35tan
4 4
5
t = =
Definition
Coterminal
Angles:
Angles
that have
the same
terminal
side.
x
y
y
x
Important Idea
To find coterminal angles,
simply add or subtract
either 360° or 2 radians
to the given angle or any
angle ...
Example
Find an
angle
coterminal
with 420°.
Find
sin420°
and
cos420°
1. Find smallest
positive
coterminal angle.
3. Apply
...
Example
Find an
angle
coterminal
1. Find smallest
positive
coterminal angle.
3. Apply
definition of sin
and cos.
Procedure...
Important Idea
The trig ratios of a given
angle and all its coterminal
angles are the same.
Try This
Find an angle that is
coterminal with 780°. Find
sin780°and cos780°.
3
sin780 sin60
2
° = ° =
1
cos780 cos60
2
° ...
Try This
Find an angle that is
coterminal with . Find
and .
sin( 10 ) sin0 0π− = =
cos( 10 ) cos0 1π− = =
10π−
sin( 10 )π−...
Important Idea
In addition to finding trig
ratios of angles ( ), we can
also find trig ratios of real
numbers in radians (...
Important Idea
There are times when we
must be satisfied with
approximate values of trig
ratios. At other times, we
can fi...
Example
cos( 2.56)−
Find the approximate value:
Since the
degree symbol
(°) is not used,
this must be
radians.
mode
Try This
Use your calculator
in radian mode to
approximate the
sin, cos and tan.
Round to 4 decimal
places. Use the
signs ...
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 pa...
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 pa...
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.
Important Idea
cos costθ = =
The definitions of the trig
ratios are the same for
angles and radians, for
example:
sin sint...
Example
Find
the
exact
value:
cos45° 45°
cos
4
π 
 ÷
 
10
10
4
π
10
10
Example
Find
the
exact
value:
sin30° 1
sin
6
π 
 ÷
 
3
1
6
π
3
30°
Definition
Reference Angle: the
angle between a given
angle and the nearest x
axis. (Note: x axis; not y
axis). Reference ...
Important Idea
How you find the reference
angle depends on which
quadrant contains the given
angle.
The ability to quickly...
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...
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 ...
Example
Find the reference angle if
the given angle is 120°.
For given
angles in quad.
2, the ref. angle
is 180° less the
...
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 ...
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 th...
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
give...
Try This
Find the reference angle if
the given angle is
x
y 4
π
−
Hint: Don’t
forget the
definition.
4
π
Important Idea
The trig ratio of a given
angle is the same as the trig
ratio of its reference angle
except, possibly, for ...
Example
Find the exact
value of the
sin, cos and tan
of the given
angle in
standard
position. Do
not use a
calculator.
135°
Procedure
1.Sketch the given angle.
2.Find and sketch the
reference angle. Label the
sides using special angle
facts.
3.Fi...
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...
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°
2
Solution
60°
3
1
3
sin60
2
° =
1
cos60
2
° =
tan60 3° =
Important Idea
x or y can be positive or
negative depending on
the quadrant but the
hypotenuse ( r ) is
always positive.
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...
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 
= − = − ÷
 
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
...
Solution 4
3
π
-1
23−
4 3
sin
3 2
π 
= − ÷
 
4 1
cos
3 2
π 
= − ÷
 
4
tan 3
3
π 
= ÷
 
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
...
The unit
circle
-1 1
-1
1
1
Definition
(1,0)
(0,1)
(-1,0)
(0,-1)
x y
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
...
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
...
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
-1 1
-1
1
Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Example
sinθ
cosθ
tanθ
csc...
-1 1
-1
1Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Example
θ
sinθ
cosθ
tanθ
cs...
-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
...
-1 1
-1
1
Find the
values of
the six trig
functions of
the given
angle in
standard
position.
2π
Example
sinθ
cosθ
tanθ
csc...
-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
cs...
-1 1
-1
1Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Example
sinθ
cosθ
tanθ
cscθ...
-1 1
-1
1Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Example
sinθ
cosθ
tanθ
cscθ...
-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...
-1 1
-1
1Find the
values of
the 6 trig
functions of
the
quadrantal
angle in
standard
position:
Try This
sinθ
cosθ
tanθ
csc...
Important Ideas
•Trig functions of quadrantal
angles have exact values.
•Trig functions of all other
angles have approxima...
Example
Use a calculator to
approximate cos 710° to 4
decimal places.
Don’t forget to check
“Mode”.
Example
Use a calculator to
approximate sin(72°30’30”)
to 4 decimal places.
Example
Use a calculator to
approximate csc 15° to 4
decimal places.
Lesson Close
How do you evaluate the
trig ratios of quadrantal
angles?
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Hprec6 4

  1. 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. 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. 3. Example Findsin ,cosθ θ & when the terminal side of the angle passes through (3,4) tanθ θ (3,4) (6,8)
  4. 4. Try This Findsin ,cosθ θ & when the terminal side of the angle passes through (6,8) tanθ θ (3,4) (6,8)
  5. 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. 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. 7. Find sin, cos & tan of the angle whose terminal side passes through the point (5,-12) θ Try This θ (5,-12)
  8. 8. Solution θ 5 -12 13 12 sin 13 θ = − 5 cos 13 θ = 12 tan 5 θ = − (5,-12)
  9. 9. Important Idea Trig ratios may be positive or negative
  10. 10. Find sin, cos & tan of the angle whose terminal side passes through the point (-5,-5) θ Try This θ (-5,-5)
  11. 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. 12. Find sin, cos & tan of the angle whose terminal side passes through the point (5,-12) θ Try This θ (5,-12)
  13. 13. Solution θ 5 -12 13 12 sin 13 θ = − 5 cos 13 θ = 12 tan 5 θ = − (5,-12)
  14. 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. 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. 16. Try This sintFind , cost when the terminal side of an angle passes through tant& on the unit circle. 3 4 , 5 5    ÷  
  17. 17. Solution 4 sin 5 t = 3 cos 5 t = 3 35tan 4 4 5 t = =
  18. 18. Definition Coterminal Angles: Angles that have the same terminal side. x y y x
  19. 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. 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. 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. 22. Important Idea The trig ratios of a given angle and all its coterminal angles are the same.
  23. 23. Try This Find an angle that is coterminal with 780°. Find sin780°and cos780°. 3 sin780 sin60 2 ° = ° = 1 cos780 cos60 2 ° = ° =
  24. 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. 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. 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. 27. Example cos( 2.56)− Find the approximate value: Since the degree symbol (°) is not used, this must be radians. mode
  28. 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. 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. 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. 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. 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. 33. Example Find the exact value: cos45° 45° cos 4 π   ÷   10 10 4 π 10 10
  34. 34. Example Find the exact value: sin30° 1 sin 6 π   ÷   3 1 6 π 3 30°
  35. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 49. 2 Solution 60° 3 1 3 sin60 2 ° = 1 cos60 2 ° = tan60 3° =
  50. 50. Important Idea x or y can be positive or negative depending on the quadrant but the hypotenuse ( r ) is always positive.
  51. 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. 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. 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. 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. 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. 56. The unit circle -1 1 -1 1 1 Definition (1,0) (0,1) (-1,0) (0,-1) x y
  57. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 70. Example Use a calculator to approximate cos 710° to 4 decimal places. Don’t forget to check “Mode”.
  71. 71. Example Use a calculator to approximate sin(72°30’30”) to 4 decimal places.
  72. 72. Example Use a calculator to approximate csc 15° to 4 decimal places.
  73. 73. Lesson Close How do you evaluate the trig ratios of quadrantal angles?

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