Transcript x - Lindsay ISD

Trigonometric Functions of
General Angles
Trigonometric Functions of General Angles
What You Will Learn:
– Find the values of trigonometric functions for
general angles.
– The definition of a reference angle.
– The signs of trigonometric functions in different
quadrants.
3 April 2016
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
2
Trigonometric Functions of General Angles
Trigonometric Representation of a General Angle
y-axis
(x, y)
y
r

x
y
sin  
r
r
csc  
y
3 April 2016
 is in the Standard Position
x-axis
x
cos  
r
r
sec  
x
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
y
tan  
x
x
cot  
y
3
Trigonometric Functions of General Angles
What are the six trig functions for this angle ?
What is r?
y
r x  y
2

5
x
–12
r
(5, –12)
3 April 2016
2
2
r x  y
2
2
r  52  122
r  13
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
4
Trigonometric Functions of General Angles
What are the six trig functions for this angle ?
y

5
x
–12
13
(5, –12)
3 April 2016
12
13
12
tan  
5
13
csc 
12
sin  
cos 
5
13
13
sec 
5
5
cot  
12
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
5
Trigonometric Functions of General Angles
Definition - Given an angle in Standard Position the Reference Angle
is the angle formed by the terminal side and the x-axis. It
is generally labeled as .
y-axis
y-axis


x-axis
x-axis
y-axis

3 April 2016
y-axis
x-axis
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook

x-axis
6
Trigonometric Functions of General Angles
y
Whatisisthe
theupper
sign
What
the sine,
rightofcorner
of the
cosine,
and
Cartesian
tangent
functions
Coordinate
inSystem
this quadrant?
labeled
as?
3 April 2016
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
Quadrant I
sine +
cosine +
tangent +
Direction of a
Positive Angle in
Standard Position
x
7
Trigonometric Functions of General Angles
Quadrant II
sine +
cosine –
y
tangent –
x
Direction of a
Negative Angle in
Standard Position
3 April 2016
Whatisisthe
theupper
sign
What
the sine,
leftofcorner
of the
cosine,
and
Cartesian
tangent
functions
Coordinate
inSystem
this quadrant?
labeled
as?
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
8
Trigonometric Functions of General Angles
y
Quadrant III
sine –
cosine –
Whatisisthe
thelower
sign
What
the sine,
leftofcorner
of the
cosine,
and
Cartesian
tangent
functions
Coordinate
inSystem
this quadrant?
labeled
as?
x
tangent +
3 April 2016
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
9
Trigonometric Functions of General Angles
Whatisisthe
thelower
sign
What
the sine,
rightofcorner
of the
cosine,
and
Cartesian
tangent
functions
Coordinate
inSystem
this quadrant?
labeled
as?
y
Quadrant IV
sine –
cosine +
x
tangent –
3 April 2016
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
10
Trigonometric Functions of General Angles
Quadrant II
sine +
cosine –
y
y
tangent –
x
Direction of a
Negative Angle in
Standard Position
Note that each trigonometric
function is positive in two
quadrants and negative in two
y
quadrants. Is there any
pattern to the sign values?
Quadrant III
sine –
cosine –
sine +
cosine +
tangent +
Direction of a
Positive Angle in
Standard Position
x
y
Quadrant IV
x
sine –
cosine +
x
tangent –
tangent +
3 April 2016
Quadrant I
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
11
Trigonometric Functions of General Angles
What You Have Learned:
– How to determine the values of trigonometric
functions for general angles.
– The definition of a reference angle and its
relationship to the original angle.
– How to determine the signs of trigonometric
functions in different quadrants.
3 April 2016
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
12
Trigonometric Functions of General Angles
END OF LINE
3 April 2016
Alg2_13_03_GeneralAngleMeasure.ppt
Copyrighted © by T. Darrel Westbrook
13