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Mathematics

MTH 108

Lun- Yi Tsai

Fall

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MTH 108
Precalculus Mathematics II
Chapter 5 Study Guide Notes
L. Sterling
Abstract
Provide a generalization to each of the key terms listed in this section.
Ray
What is a ray?
A ray, which can also be classiﬁed as a “half-line”, is a line that has only one end; rays start at one
given point, which can also be called an "endpoint", and then it extends to any direction forever,
which can be considered as "going to inﬁnity".
Initial and Terminal Sides
What is the diﬀerence between an initial and terminal side?
Initial Sides
▯
Initial sides are normally half-lines that are at 0 with a vertex while holding the given half-line
in place.
Terminal Sides
Terminal sides are normally rotating to determine either a positive or negative angle, which can
be explained by the following:
Counterclockwise Positive
Clockwise Negative
Angle ▯
Quadrant Angles
Lying in Quadrant I
▯ ▯
▯ must be in the ﬁrst quadrant iﬀ (if and only if) ▯ is between 0 and 90 .
Lying in Quadrant II
▯ ▯
▯ must be in the second quadrant iﬀ (if and only if) ▯ is between 90 and 180 .
Lying in Quadrant III
▯ must be in the third quadrant iﬀ (if and only if) ▯ is between 180 and 270 .
Quadrantal Angle
▯ must be the quadrantal angle iﬀ (if and only if) ▯ is exactly 270 .
1 Lying in Quadrant IV
▯ must be in the ﬁrst quadrant iﬀ (if and only if) ▯ is between 270 and 360 .
Revolutions
▯ ▯ ▯ ▯
90 180 270 360
Right Angle Straight Angle Right Angle Straight Angle
1 1 3
4Revolution 2 Revolution 4Revolution 1 Revolution
Counterclockwise Counterclockwise Counterclockwise Counterclockwise
Conversions
Degrees to Minutes to Seconds
One degree can be equivalent to 60 minutes, which is also equivalent to 3600 seconds, which can
be explained by the following:
1 = 60 = 3600"
Minutes to Degrees
1
One minute is equivalent t60 degrees, which can be explained by the following:
▯ ▯ ▯
1 = 60" = 1 ▯ 0:0167 ▯
60
Seconds to Degrees
One second is equivalent to1 degrees, which can be explained by the following:
3600
▯ ▯ ▯ ▯ ▯ ▯
1 0 1 1 ▯ 1 ▯ ▯
1 = = ▯ = ▯ 0:0002778
60 60 60 3600
Degrees, Minutes, and Seconds
When you have degrees, which is normally d, minutes, which is normally m, and seconds, which is
normally s, can be expressed by the following:
d m s"
Decimal Degrees
m s
One decimal degree, which is normally dd, is equivalent to degree60plus 3600, which can be
explained by the following:
m s
dd = d + 60 + 3600
Circle’s Arc Length
Proportional Ratio
The following is the proportional ratio for the arc length:
▯ s
=
▯1 s1
2 Arc Length
The arc length is when the arc length, which is normally labeled s, is being subtended by the
circle’s radius, which is normally labeled r, and its central angle, which is normally labeled with ▯
radians, which is all expressed by the following formulas:
s = r▯
Length s
Radius r
Central Angle ▯ radians
Central Angle
The central angle is an angle with a vertex being the circle’s center.
Conversions
Revolutions to Radians
1 Revolution = 2▯ Radians
s = 2▯r
Degrees to Radians
▯
1 = Radians
180
180 = ▯ Radians
Circle’s Sector Area
Proportional Ratio
The following is the proportional ratio for the sector area:
▯ = A
▯1 A1
Sector Area
The sector area is when half of the circle’s squared radius, which is normally la2r or either
r2
2, and its central angle, which is normally labeled with ▯ radians, which is all expressed by the
following formulas:
1 2
A = 2 r ▯
Area A
Radius r
Central Angle ▯ radians
Linear and Angular Speed
Linear Speed
The linear speed, or linear velocity, occurs when the distance, which is normally labeled s, is being
traveled in time, which is normally labeled t, around its circle, which would make its speed/velocity,
which is normally labeled v, which is all expressed by the following formula:
v = s
t
3 Angular Speed
The angular speed, or angular velocity, occurs when the angle, which is normally labeled !, is
when the radians, which is normally labeled ▯, while being divided by its elapsed time, which is
normally labeled t, which is all expressed by the following formula:
▯
! =
t
Converting Linear and Angular Speed
When it comes to linear and angular speed, the linear speed, which is normally labeled v, is equaled
to the radians, which is normally labeled r, per unit time multiplied by the angular speed, which
is normally labeled !, which is all expressed by the following formula:
v = r!
▯ in Degrees and Radians
When it comes to the value of a function, that’s trigonometric, of any possible angle would be
equaled to the exact same function, that’s also trigonometric, of an angle, which means ▯ in this
case, that it coterminal to a given angle.
Degrees : 0 ▯ ▯ ▯ 360 ▯
Radians : 0 ▯ ▯ ▯ 2▯
Thanks to the angle, which also means ▯ in this case, and the following where k can be expressed
as any integer would be coterminal:
Degrees : ▯ + 360 k
Radians : ▯ + 2▯k
Signs of Trigonometric Functi

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