Download Definition of trigonometric functions

Dr. Nestler - Math 2 - 5.2 - Definitions of the Trigonometric Functions
There are two ways to define the trigonometric functions.
1. Definition of the Trigonometric Functions at a Real Number
Suppose > is a real number. Let ÐBß CÑ be the point obtained by moving along the unit circle
from the point Ð"ß !Ñ a distance of > units counterclockwise if > ! and l>l units
clockwise if >  !. We call ÐBß CÑ the point on the unit circle corresponding to >. Note
that each number > determines a unique pair of numbers B and C in this manner.
The six trigonometric functions are defined at > as follows:
cosine:
sine:
tangent:
cos Ð>Ñ œ B
sin Ð>Ñ œ C
tan Ð>Ñ œ CB ÐB Á !Ñ
secant:
cosecant:
cotangent:
sec Ð>Ñ œ
csc Ð>Ñ œ
cot Ð>Ñ œ
"
B
"
C
B
C
ÐB Á !Ñ
ÐC Á !Ñ
ÐC Á !Ñ
So the point on the unit circle corresponding to > has coordinates Ðcos >ß sin >Ñ.
Example: Evaluate the trigonometric functions at > œ !, > œ 1 and > œ #1Þ
2. Definition of the Trigonometric Functions at an Angle
Suppose ) is an angle in standard position (with vertex at the origin and initial side along the
positive B-axis). Let ÐBß CÑ be the point of intersection of the unit circle with the terminal
side of ). As above, each angle ) determines a unique pair of numbers B and C in this
manner.
The six trigonometric functions are defined at ) as follows:
cosine:
sine:
tangent:
cos Ð)Ñ œ B
sin Ð)Ñ œ C
tan Ð)Ñ œ CB ÐB Á !Ñ
secant:
cosecant:
cotangent:
sec Ð)Ñ œ
csc Ð)Ñ œ
cot Ð)Ñ œ
"
B
"
C
B
C
ÐB Á !Ñ
ÐC Á !Ñ
ÐC Á !Ñ
3. These two definitions are compatible because:
If ) is a central angle of radian measure between 0 and #1, then the length > of the arc of the unit
circle cut out by the angle equals the radian measure of ). Since ) œ >, the values of the
trigonometric functions at ) equal the values of the trigonometric functions at >.
Example: Evaluate the trigonometric functions at ) œ
1
#
and ) œ
Example: Evaluate the trigonometric functions at ) œ 1% .
$1
# .
Example: Evaluate the trigonometric functions at ) œ
1
'
and ) œ 1$ Þ
We can use the symmetries of the circle to compute trigonometric values at many other inputs.
Definition. Suppose that ) is an angle in standard position with terminal side not on the axes.
The reference angle corresponding to ) is the acute angle between the B-axis and the
terminal side of ).
By the symmetries of the circle, the values of the trigonometric functions at an angle ) are the
same as the values of the trigonometric functions at its reference angle, or are off by a
sign.
Example: Compute the trigonometric values at ) œ  #$1 .
To evaluate a trigonometric function at a number/angle:
(1) Evaluate the function at the reference angle.
(2) Use ASTC to determine the sign.
We must be able to quickly find the values of the trigonometric functions at all integer multiples
of
1
%
and 1' , where they are defined.
[Can do 5.2 #7-64, 93-106, 117, 118]