Download 2.1 - UCR Math Dept.

Math 36 "Fall ’08"
2.1 "Angles in the Cartesian Plane"
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Skills Objectives:
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Plot angles in standard position
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Identify coterminal angles
*
Graph common angles
Conceptual Objectives:
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Relate the x and y coordinate to the legs of a right triangle
*
Derive the distance formula from the Pythagorean theorem
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Preliminaries:
We already know that:
-
A common unit of measure for angles is degrees.
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The sum of the measure of the three angles of a triangle is 180 :
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How to use the Pythagorean theorem (which relates the lengths of the tree sides of a right triangle).
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Counterclockwise =) positive angles.
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Clockwise =) negative angles.
Based on what we learned, the following two angles have same measurement of 90 :
We need a frame of reference. In this section we use the Cartesian plane as our frame of reference.
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Angles in Standard Position
De…nition:
"Standard Position"
An angle is said to be in standard position if its _________________________
is along the _____________________ and its vertex is at the ___________.
Note:
We say that an angle lies in the quadrant in which its terminal side lies.
Example 1: (Sketching angles in standard position and determining where the angle lies)
Sketch the following angles in standard position, and state the quadrant in which the terminal side lies:
a)
Acute angle
c)
Angles with measure 180 <
De…nition:
< 270
b)
Obtuse angle
d)
Angles with measure 270 <
"Quadrantal Angles"
Angles in standard position with terminal side along the x
axis or y
(_____________________) are called quadrantal angles.
If we put everything together we get:
2
axis
< 360
Example 2: (Sketching angles in standard position and determining where the angle lies)
Sketch the following angles in standard position, and state the quadrant in (or axis on) which the terminal side lies:
a)
300
b)
540
Coterminal Angles
De…nition:
"Coterminal Angles’
Two angles in standard position with the ___________________________
are called coterminal angles.
NOTE:
Moreover, all coterminal angles have also the same initial side (positive x
axis).
Example 3: (Recognizing coterminal angles)
Determine if the following pairs of angles are coterminal:
a)
= 240
b)
= 20
Question:
and
and
=
=
120
380
Is there a more direct way to determine whether two angles are coterminal ?
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Method 1:
Finding Measures of Coterminal Angles
If the given angle is positive, __________________ repeatedly until the result is
a positive angle less than or equal to 360 :
If the given angle is negative, _________________ repeatedly until the result is
a positive angle less than or equal to 360 :
Method 2:
Finding Measures of Coterminal Angles
kUse the following equation _____________________ where k is an integer.k
Example 4: (Finding measures of coterminal angles)
Find the smallest possible positive angle that are coterminal with the following angles:
a)
900
b)
430
Common Angles in Standard Position
The common angles for which we determined exact values for trigonometric functions in Chapter 1 are 30 ; 45 ; and 60 :
Recall the relationships between these triangles.
30
60
90
45
45
90
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If we let the hypotenuse equal to one, then we have the following triangles:
We can position these triangles on the Cartesian plane, so that we have three angles (30 ; 45 ; and 60 )
in standard position.
Notice that the x
coordinate and y
y
coordinate correspond to the side lengths.
y
y
x
x
If we graph the three angles on the same Cartesian coordinate system, we get the following in the …rst quadrant:
y
x
5
x
Indeed we can graph the other three quadrants by using symmetry.
Notice that all of these coordinate pairs satisfy the equation of a unit circle (x2 + y 2 = 1)
y
x
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