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General Trigonometric Functions
Ó 2006, Frank Ma
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General Trigonometric Functions
Objective:
*  Definition of trigonometric functions
    for angles of any measurements.
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Standard Position of an Angl= e
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Standard Position of an Angl= e
When an angle is set in <= /span>
the xy-coordinate =
system with the tip at
the origin and a fixed = side coincides with the <= /span>
x-axis, we say the =
angle is in the
standard position= .
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Positive and Negative Angles
Dialing for angles:
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Positive and Negative Angles
Dialing for angles:
Its useful to think of angles = are produced by dialing
a dial against the x-axis. 
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Positive and Negative Angles
Dialing for angles:
Its useful to think of angles = are produced by dialing
a dial against the x-axis.  The two clock-directions
for the dial determine the sig= ns of the angles.
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•An angle opens <= /span>
•counter-clock wisely&= #13;
•has positive measurem= ent.
Positive and Negative Angles
Dialing for angles:
Its useful to think of angles = are produced by dialing
a dial against the x-axis.  The two clock-directions
for the dial determine the sig= ns of the angles.
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•An angle opens <= /span>
•counter-clockwisely&#= 13;
•has positive measurem= ent.
Positive and Negative Angles
Dialing for angles:
Its useful to think of angles = are produced by dialing
a dial against the x-axis.  The two clock-directions
for the dial determine the sig= ns of the angles.
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•An angle opens <= /span>
•counter-clockwisely&#= 13;
•has positive measurement.
•
•An angle opens
•clockwisely has neg= ative
•measurement.=
Positive and Negative Angles
Dialing for angles:
Its useful to think of angles = are produced by dialing
a dial against the x-axis.  The two clock-directions
for the dial determine the sig= ns of the angles.
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•An angle opens <= /span>
•counter-clockwisely&#= 13;
•has positive measurement.
•
•An angle opens
•clockwisely has neg= ative
•measurement.=
Positive and Negative Angles
Dialing for angles:
Its useful to think of angles = are produced by dialing
a dial against the x-axis.  The two clock-directions
for the dial determine the sig= ns of the angles.
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Dial= ing for Angles
We may dial an angle of any degree, positive or negative.<= /span>
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Dial= ing for Angles
We may dial an angle of any degree, positive or negative.<= /span>
Example A: Draw the angle 560o and -1120o
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Dial= ing for Angles
We may dial an angle of any degree, positive or negative.<= /span>
For every ±360o, the dial makes one complete round and returns = to the starting position. <= /div>
Example A: Draw the angle 560o and -1120o
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Dial= ing for Angles
We may dial an angle of any degree, positive or negative.<= /span>
For every ±360o, the dial makes one complete round and returns = to the starting position. <= /div>
Example A: Draw the angle 560o and -1120o
560÷360 =3D 1 with remainder 200, hen= ce the dial stops at the same position as 200o.&nb= sp;
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Dial= ing for Angles
We may dial an angle of any degree, positive or negative.<= /span>
For every ±360o, the dial makes one complete round and returns = to the starting position. <= /div>
Example A: Draw the angle 560o and -1120o
560÷360 =3D 1 with remainder 200, hen= ce the dial stops at the same position as 200o.&nb= sp;
560o, 200o
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Dial= ing for Angles
We may dial an angle of any degree, positive or negative.<= /span>
For every ±360o, the dial makes one complete round and returns = to the starting position. <= /div>
Example A: Draw the angle 560o and -1120o
560÷360 =3D 1 with remainder 200, hen= ce the dial stops at the same position as 200o.&nb= sp;
-1120 =3D 3*(-360) + (-40).
560o, 200o
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Dial= ing for Angles
We may dial an angle of any degree, positive or negative.<= /span>
For every ±360o, the dial makes one complete round and returns = to the starting position. <= /div>
Example A: Draw the angle 560o and -1120o
560÷360 =3D 1 with remainder 200, hen= ce the dial stops at the same position as 200o.&nb= sp;
-1120 =3D 3*(-360) + (-40). = Hence the dial stops at the same position as -40o.&nb= sp;
560o, 200o
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Dial= ing for Angles
We may dial an angle of any degree, positive or negative.<= /span>
For every ±360o, the dial makes one complete round and returns = to the starting position. <= /div>
Example A: Draw the angle 560o and -1120o
560÷360 =3D 1 with remainder 200, hen= ce the dial stops at the same position as 200o.&nb= sp;
-1120 =3D 3*(-360) + (-40). = Hence the dial stops at the same position as -40o.&nb= sp;
560o, 200o
-1120o, -40o
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G= eneral Trigonometric Functions
 (Definition for angles of all measurements)
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Given angle A in the standard position as shown,
G= eneral Trigonometric Functions
 (Definition for angles of all measurements)
A
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length of the dial =3D r
= Given angle A in the standard position as shown,
= let r =3D length of the dial (hence r > 0),
G= eneral Trigonometric Functions
 (Definition for angles of all measurements)
A
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length of the dial =3D r
= Given angle A in the standard position as shown,
= let r =3D length of the dial (hence r > 0),
= let (x, y) be the coordinate of the dial-tip,
<= b>tip of the dial =3D (x<= /span>, y)
G= eneral Trigonometric Functions
 (Definition for angles of all measurements)
A
x
y
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length of the dial =3D r
= Given angle A in the standard position as shown,
= let r =3D length of the dial (hence r > 0),
= let (x, y) be the coordinate of the dial-tip, we define:
sin(A) =3D
y
r&#= 13;
<= b>tip of the dial =3D (x<= /span>, y)
G= eneral Trigonometric Functions
 (Definition for angles of all measurements)
A
x
y
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length of the dial =3D r
= Given angle A in the standard position as shown,
= let r =3D length of the dial (hence r > 0),
= let (x, y) be the coordinate of the dial-tip, we define:
sin(A) =3D
y
r&#= 13;
cos(A) =3D
x
r&#= 13;
<= b>tip of the dial =3D (x<= /span>, y)
G= eneral Trigonometric Functions
 (Definition for angles of all measurements)
A
x
y
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length of the dial =3D r
= Given angle A in the standard position as shown,
= let r =3D length of the dial (hence r > 0),
= let (x, y) be the coordinate of the dial-tip, we define:
sin(A) =3D
y
r&#= 13;
cos(A) =3D
x
r&#= 13;
tan(A) =3D
y
x
<= b>tip of the dial =3D (x<= /span>, y)
G= eneral Trigonometric Functions
 (Definition for angles of all measurements)
A
x
y
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length of the dial =3D r
= Given angle A in the standard position as shown,
= let r =3D length of the dial (hence r > 0),
= let (x, y) be the coordinate of the dial-tip, we define:
sin(A) =3D
y
r&#= 13;
cos(A) =3D
x
r&#= 13;
tan(A) =3D
y
x
<= b>tip of the dial =3D (x<= /span>, y)
G= eneral Trigonometric Functions
 (Definition for angles of all measurements)
A
x
y
Note: If 0o < = A < 90o, these definitions coincide with the SOHCAHTOA definition via right triangles.
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The Trig-values are independent of the lengths of the dials in the definition. 
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The Trig-values are independent of the lengths of the dials in the definition. 
Suppose we have two dials with lengths of
r and R.
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The Trig-values are independent of the lengths of the dials in the definition. 
Suppose we have two dials with lengths of
r and R.
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The Trig-values are independent of the lengths of the dials in the definition. 
Suppose we have two dials with lengths of
r and R. From facts of = similar triangles, the = corresponding sides are proportional.
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The Trig-values are independent of the lengths of the dials in the definition. 
Suppose we have two dials with lengths of
r and R. From facts of = similar triangles, the = corresponding sides are proportional. <= /span>
sin(A) =3D
y
r
cos(A) =3D
x
r
tan(A) =3D
y
x
=3D&#= 13;
Y&#= 13;
R&#= 13;
=3D&#= 13;
X&#= 13;
R&#= 13;
=3D&#= 13;
Y&#= 13;
X&#= 13;
r
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Basic = Facts of SCT
A
sin(A) =3D
y
r
cos(A) =3D
x
r
tan(A) =3D
y
x
Tip of the
dial at (x= , y)
length= of the dial =3D r
x
y
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Basic = Facts of SCT
A
•  r2 =3D x2 + y2 by the Pythagorean theor= em
sin(A) =3D
y
r
cos(A) =3D
x
r
tan(A) =3D
y
x
Tip of the
dial at (x= , y)
length= of the dial =3D r
x
y
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Basic = Facts of SCT
A
•  r2 =3D x2 + y2 by the Pythagorean theor= em
• -1 < sin(A) < 1 since | y | < r
sin(A) =3D
y
r
cos(A) =3D
x
r
tan(A) =3D
y
x
Tip of the
dial at (x= , y)
length= of the dial =3D r
x
y
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Basic = Facts of SCT
A
•  r2 =3D x2 + y2 by the Pythagorean theor= em
• -1 < sin(A) < 1 since | y | < r
• -1 < cos(A) < 1 since | x | < r
sin(A) =3D
y
r
cos(A) =3D
x
r
tan(A) =3D
y
x
Tip of the
dial at (x= , y)
length= of the dial =3D r
x
y
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Basic = Facts of SCT
A
•  r2 =3D x2 + y2 by the Pythagorean theor= em
• -1 < sin(A) < 1 since | y | < r
• -1 < cos(A) < 1 since | x | < r
• -∞ < tan(A) &l= t;∞ an= d tan(±90), tan(±270= ), .. are undefined because x =3D 0.
sin(A) =3D
y
r
cos(A) =3D
x
r
tan(A) =3D
y
x
Tip of the
dial at (x= , y)
length= of the dial =3D r
x
y
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Example B:  Given the = angle A as shown, find its trig-values S= CT.
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Use Pythagorean Theorem.
to find r first.
Example B:  Given the = angle A as shown, find its trig-values S= CT.
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Use Pythagorean Theorem.
to find r first.
r2 =3D (-3)2 + 42
Example B:  Given the = angle A as shown, find its trig-values S= CT.
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Use Pythagorean Theorem.
to find r first.
r2 =3D (-3)2 + 42 <= /span>
r2  =3D 25
r =3D 5
Example B:  Given the = angle A as shown, find its trig-values S= CT.
r =3D 5
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Use Pythagorean Theorem.
to find r first.
r2 =3D (-3)2 + 42 <= /span>
r2  =3D 25
r =3D 5
Example B:  Given the = angle A as shown, find its trig-values S= CT.
sin(A) =3D=
y
r
cos(A) =3D=
x
r
tan(A) =3D=
y
x
=3D=
=3D=
=3D=
Hence
r =3D 5
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