Transcript Angle - MrGranquistsTrigonometryPage

1.1
Trigonometry
Vocabulary:
Angle – created by rotating a ray about its endpoint.
Initial Side – the starting position of the ray.
Terminal Side – the position of the ray after rotation.
Vertex – the endpoint of the ray.
This arrow means
that the rotation was
in a
counterclockwise
direction.
Vertex
Initial side
This arrow
means that
the
rotation
was in a
clockwise
direction.
Vertex
Terminal side
Positive Angles – angles generated by a counterclockwise
rotation.
Negative Angles – angles generated by a clockwise
rotation.
We label angles in trigonometry by using the Greek
alphabet.
 - Greek letter alpha
 - Greek letter beta
 - Greek letter phi
 - Greek letter theta
This represents a
positive angle
Vertex
Initial side
This
represents
a negative
angle
Vertex
Terminal side
Standard Position – an angle is in standard position
when its initial side rests on the positive half of the
x-axis.
Positive angle in standard position
Practice sketching graphs in standard position: (degrees only)
There are two ways to measure angles…
Degrees
Radians
Degrees:
• There are 360 in a complete circle.
• 1 is 1/360th of a rotation.
Radians:
• There are 2 radians in a complete circle.
• 1 radian is the size of the central angle when the
radius of the circle is the same size as the arc of
the central angle.
Length of the
arc is equal to
the length of
the radius.
arc
1 Radian
radius
Practice sketching graphs in standard positions with radians:
Coterminal angles – two angles that share a common
vertex, a common initial side and a common terminal
side.
Examples of Coterminal Angles


 and  are coterminal
angles because they share
the same initial side and
same terminal side.
Coterminal angles could
go in opposite directions.
Examples of Coterminal Angles


 and  are coterminal
angles because they share
the same initial side and
same terminal side.
Coterminal angles could
go in the same direction
with multiple rotations.
Finding coterminal angles of angles measured in
degrees:
Since a complete circle has a total of 360, you can
find coterminal angles by adding or subtracting 360
from the angle that is provided.
Example:
Find two coterminal angles (one positive and one
negative) for the following angles.
 = 25
positive coterminal angle:
25 + 360 = 385
negative coterminal angle:
25 – 360 = - 335
Example:
Find two coterminal angles (one positive and one
negative) for the following angles.
 = 725
positive coterminal angle:
725 + 360 = 1085 (add a rotation)
or
725 – 360 = 365 (subtract a rotation)
or
725 – 360 – 360 = 5 (subtract 2 rotations)
negative coterminal angle:
725 – 360 – 360 – 360 = - 355
(must subtract 3 rotations)
Example:
Find two coterminal angles (one positive and one
negative) for the following angles.
 = -90
positive coterminal angle:
-90 + 360 = 270
negative coterminal angle:
- 90 – 360 = - 450
Finding coterminal angles of angles measured in
radians:
Since a complete circle has a total of 2 radians you
can find coterminal angles by adding or subtracting
2 from the angle that is provided.
Example:
Find two coterminal angles (one positive and one
negative) for the following angles.
 = /7
positive coterminal angle:
/
7
+ 2 = /7 +
14/ = 15/ rad
7
7
negative coterminal angle:
/
7
- 2 = /7 -
14/
7=
-13/ rad
7
Example:
Find two coterminal angles (one positive and one
negative) for the following angles.
=
-4/
9
positive coterminal angle:
-4/
9
+2 =
-4/
9
+
18/ =14/ rad
9
9
negative coterminal angle:
-4/
9
-2 =-4/9 -
18/ =-22/ rad
9
9
Complementary angles – two positive angles whose
sum is 90 or two positive angles whose sum is /2.
To find the complement of a given angle you
subtract the given angle from 90 (if the angle
provided is in degrees) or from /2 (if the angle
provided is in radians).
.
Example:
Find the complement of the following angles if one
exists.
 = 29
complement = 90 – 29 = 61
 = 107
complement = 90 – 107 = none
(No complement because it is negative)
 =  /5
complement = /2 - /5 =
5/
2/
10
10
=
3/
10
Supplementary angles – two positive angles whose
sum is 180 or two positive angles whose sum is .
To find the supplement of a given angle you subtract
the given angle from 180 (if the angle provided is in
degrees) or from  (if the angle provided is in
radians).
Example:
Find the supplement of the following angles if one
exists.
 = 29
supplement = 180 – 29 = 151
 = 107
supplement = 180 – 107 = 73
 =  /5
supplement = - /5 =
5/
5
- /5 =
4/5
We have to become comfortable
working with both forms of measuring
angles.
Therefore, MEMORIZE the following:
Degrees
0
30
45
60
Radians
0 radians
/6 radians
/4 radians
/3 radians
Degrees
90
180
270
360
Radians
/2 radians
 radians
3/2 radians
2 radians
We will memorize more, very, very soon.
Manually Converting from Degrees to Radians:
Multiply the given degrees by  radians/180
Example:
Convert the following degrees to radians
135
135 degrees
 radians
1
180 degrees
135 radians
180
3 radians
4
=
=
Multiply the given degrees by  radians/180
Example:
Convert the following degrees to radians
540
540 degrees
 radians
1
180 degrees
540 radians
180
3 radians
1
=
=
Manually Converting from Radians to Degrees:
Multiply the given radians by 180/ radians
Example:
Convert the following radians to degrees.
-/3 radians
- radians
3
180 degrees
 radians
-180 degrees
3
-60
=
=
Multiply the given radians by 180/ radians
Example:
Convert the following radians to degrees.
9/2 radians
9 radians
2
180 degrees
 radians
1620 degrees
2
810
=
=
Multiply the given radians by 180/ radians
Example:
Convert the following radians to degrees.
2
(if you don’t see the degree
symbol, then the angle
measure is automatically
believed to be a radian.)
2 radians
1
360
2
180 degrees
 radians
degrees
114.59
=
=
Tomorrow, we can look at your individual calculators
and show you how to do these conversions via those
calculators.
Finding Arc Length:
•The following formula is used to
determine arc length:
s=r
arc length
radius
must have the same units
of measure
Measure of the
central angle in
radians.
Examples
s=?
r= 14 inches
s=r
s = (14)(3)
s = 42 inches
Picture not drawn to scale.
Examples
s =9 cm
You must convert
30 to radians.
r= ?
s=r
9 = (r)(/6)
r = 54/  cm  17.19 cm
Picture not drawn to scale.