Download 2.2 - UCR Math Dept.

Math 36 "Fall ’08"
2.2 "De…nition 2 of Trigonometric Functions: Cartesian Plane"
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Skills Objectives:
*
Calculate trigonometric function values for acute angles
*
Calculate trigonometric function values for nonacute angles
*
Calculate trigonometric function values for quadrantal angles
Conceptual Objectives:
*
De…ne trigonometric functions in the Cartesian plane
*
Understand why some trigonometric functions are unde…ned for quadrantal angles
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Preliminaries:
In chapter one we de…ned trigonometric functions as ratios of side lengths of right triangles.
In chapter two, we will de…ne trigonometric functions as ratios of x and y coordinates and distance in the Cartesian
plane, which is consistent with right triangle trigonometry for acute angles. However, this second approach enables
us to formulate trigonometric functions for quadrantal angles and nonacute angles (i:e: 90 ; 180 ; 270 ; 360 ).
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Trigonometric Functions: Cartesian Plane
To de…ne the trigonometric functions in the Cartesian plane, we start with an acute angle ( ) ; in standard position.
Choose any point (x; y) on the terminal side of the angle as long as it is not the vertex (origin).
y
y
x
x
1
De…nition (2):
Trigonometric Functions
Let (x; y) be a point other that the origin on the terminal side of an angle
in standard position.
Let r be the distance from the point (x; y) to the origin; then the six trigonometric functions are de…ned as:
Example 1: (Calculating trigonometric function values for acute angles)
The terminal side of an angle ( ) in standard position passes through the point (3; 7).
Calculate the values of the six trigonometric functions for angle .
y
8
6
4
2
-2
2
4
x
-2
Example 2: (Calculating trigonometric functions values for nonacute angles)
The terminal side of an angle ( ) in standard position passes through the point ( 3; 5).
Calculate the values of the six trigonometric functions for angle .
y
-4
2
-2
2
-2
x
-4
-6
-8
2
Question:
If we say that the terminal side of an angle is given by an equation of a line that passes through the origin,
Can we determine the angle?
Answer:
Question:
Once we know which part of the line represents the terminal side of the angle,
Does it matter which point on the line we use to formulate the trigonometric function values?
Answer:
Example 3: (Calculating trigonometric function values for nonacute angles)
Calculate the values for the six trigonometric functions of the angle ( ) ; given in standard position,
if the terminal side of
is de…ned by y = 2x, x
0:
y
4
2
-4
-2
2
-2
4
x
-4
3
So far, we have avoided evaluating expressions such as sin 90 because 90 is not an acute angle. However, with our
second de…nition of trigonometric function in the Cartesian plane we now are able to formulate the trigonometric
functions for quadrantal angles.
Example 4: (Calculating trigonometric function values for quadrantal angles)
Calculate the values for the six trigonometric functions when
= 90
Example 5: (Calculating trigonometric function values for quadrantal angles)
Calculate the values for the six trigonometric functions when
= 180
Example 6: (Calculating trigonometric function values for quadrantal angles)
Calculate the values for the six trigonometric functions when
4
= 270
Example 7: (Calculating trigonometric function values for quadrantal angles)
Calculate the values for the six trigonometric functions when
= 360
The following table summarizes the trigonometric function values for common quadrantal angles.
sin
cos
tan
sec
0
90
180
270
360
5
csc
cot