ANGLES IN A UNIT CIRCLE || PRE-CALCULUS
Summary
TLDRThis educational video lesson delves into the concept of angles in a unit circle, explaining the geometric definition and its measurement in degrees. It covers the impact of direction of rotation on angle measurement, distinguishing between clockwise and counter-clockwise rotations. The video also instructs viewers on converting angle measures from decimal degrees to degrees, minutes, and seconds, and vice versa, using practical examples to clarify the process.
Takeaways
- 📐 Angles in a unit circle are discussed with a focus on geometry and trigonometry perspectives.
- 🔄 An angle is defined as the amount of rotation needed to move from the initial side to the terminal side.
- 👉 The direction of rotation affects angle measurement, with counter-clockwise being positive and clockwise being negative.
- 🌀 A full rotation (360 degrees) is equivalent to 0 degrees, illustrating the cyclical nature of angles.
- 🔢 The degree measure system, developed by Babylonians around 2000 BC, uses a sexagesimal (base-60) numeral system.
- ⏱️ One degree is equivalent to 60 minutes, and one minute is equivalent to 60 seconds.
- 📉 The script provides examples of converting decimal degrees to degrees, minutes, and seconds.
- 📈 It also demonstrates converting degrees, minutes, and seconds back to decimal degrees.
- 🔄 The process of converting between these units involves multiplying by 60 to go from degrees to minutes or from minutes to seconds.
- 📚 The tutorial is educational, aiming to teach viewers how to work with angles and their measurements.
- 🎓 The video concludes with a prompt for viewers to like, subscribe, and hit the bell button for more educational content.
Q & A
What is the definition of an angle in geometry?
-An angle is defined as the union of two non-collinear rays which have the same endpoint.
How does the definition of an angle in trigonometry differ from the geometric definition?
-In trigonometry, an angle is viewed as the amount of rotation generated when a ray is rotated about its endpoint, and it requires labeling the sides of the angle as initial and terminal.
What are the initial and terminal sides of an angle?
-The initial side is the position of the ray at the start, and the terminal side is the position of the ray at the end of its rotation.
How does the direction of rotation affect angle measures?
-Counter-clockwise rotation results in positive angle measures, while clockwise rotation results in negative angle measures.
What is the significance of a full rotation in terms of degrees?
-A full rotation is equivalent to 360 degrees.
How are degrees, minutes, and seconds related to each other in angle measurement?
-One degree is equal to 60 minutes, and one minute is equal to 60 seconds.
How can you convert a decimal degree measure to minutes and seconds?
-First, separate the whole number of degrees from the decimal. Then, multiply the decimal part by 60 to convert it to minutes. If there's a decimal in the minutes, multiply that decimal by 60 to convert it to seconds.
What is the final result when converting 22.4 degrees to degrees, minutes, and seconds?
-The final result is 22 degrees 24 minutes.
How do you convert 14.21 degrees to degrees, minutes, and seconds?
-First, convert the 0.21 degrees to minutes by multiplying by 60, resulting in 12.6 minutes. Then, take the decimal part of the minutes (0.6) and convert it to seconds by multiplying by 60, resulting in 36 seconds. The final answer is 14 degrees 12 minutes 36 seconds.
How can you convert a degree measurement in minutes and seconds to decimal degrees?
-Convert the minutes to a fraction of a degree by dividing by 60, and the seconds to a fraction of a degree by dividing by 3600 (60 seconds per minute and 60 minutes per degree). Then, add these fractions to the whole number of degrees.
What is the result when converting 31 degrees 12 minutes and 54 seconds to decimal degrees?
-First, convert the 54 seconds to minutes by dividing by 60, which gives 0.9 minutes. Then add this to the 12 minutes to get 12.9 minutes. Convert this to degrees by dividing by 60, which gives approximately 0.215 degrees. Add this to the 31 degrees to get a final result of approximately 31.215 degrees.
How is a negative angle measured in terms of degrees, minutes, and seconds?
-A negative angle is measured by converting the minutes and seconds to a decimal form and then subtracting that from the degrees. For example, -5 degrees 48 minutes 41 seconds is converted to -5.81 degrees by first converting 41 seconds to 0.68 minutes and then adding that to 48 minutes to get 48.68 minutes, which is then converted to 0.81 degrees and subtracted from -5 degrees.
Outlines
📚 Introduction to Angles in a Unit Circle
This paragraph introduces the concept of angles in a unit circle within the context of geometry and trigonometry. It explains that an angle is defined as the union of two non-collinear rays with a common endpoint, and traditionally measures between 0 and 180 degrees. However, in trigonometry, angles are considered as rotations around their endpoint, necessitating the labeling of the initial and terminal sides of the angle. The direction of rotation is also significant, with clockwise rotations resulting in negative angle measures and counterclockwise rotations in positive measures. The Babylonians' sexagesimal system is mentioned as the basis for degree measure, which equates one complete rotation to 360 degrees.
🔢 Converting Decimal Degrees to Minutes and Seconds
The second paragraph focuses on converting decimal degree measurements to minutes and seconds. It explains the sexagesimal system used by the Babylonians, where one degree is equivalent to 60 minutes, and one minute to 60 seconds. The process involves multiplying the decimal part of the degree by 60 to convert it to minutes, and if necessary, the decimal part of the minutes by 60 to convert to seconds. Examples are provided to demonstrate the conversion process, such as converting 22.4 degrees to 22 degrees and 24 minutes, and 14.21 degrees to 14 degrees, 12 minutes, and 36 seconds.
🔄 Converting Degree Measurements to Decimal Degrees
This paragraph discusses the conversion of degree measurements that include minutes and seconds to decimal degrees. It outlines the process of converting seconds to minutes by dividing by 60, and then adding that value to the minutes, which is then converted to decimal degrees by dividing by 60. Examples are given, such as converting 31 degrees 12 minutes and 54 seconds to 31.215 degrees, and 50 degrees 22 minutes and 11 seconds to 50.37 degrees. The process is also demonstrated for negative values, like converting -5 degrees 48 minutes and 41 seconds to -5.81 degrees.
📢 Conclusion and Call to Action
The final paragraph serves as a conclusion to the video tutorial, inviting viewers to like, subscribe, and hit the bell button for updates on more video tutorials. It is a call to action for the audience to engage with the content and stay connected for future educational videos on the Walmart channel.
Mindmap
Keywords
💡Angle
💡Unit Circle
💡Rotation
💡Initial Side
💡Terminal Side
💡Vertex
💡Degrees
💡Sexagesimal System
💡Conversion
💡Calculator
💡Trigonometry
Highlights
Definition of an angle in geometry as the union of two non-collinear rays with a common endpoint.
Trigonometric perspective on angles as the amount of rotation from the initial side to the terminal side.
Labeling the sides of an angle: initial side and terminal side.
The direction of rotation affects angle measures: counterclockwise is positive, clockwise is negative.
A full rotation is equivalent to 360 degrees.
Conversion of decimal degrees to minutes and seconds.
Example of converting 22.4 degrees to minutes and seconds.
Conversion of 14.21 degrees to minutes and seconds, including handling of decimal minutes.
The sexagesimal system used for degree, minute, and second measurements.
Conversion of degree measurements to decimal degrees.
Example of converting 31 degrees 12 minutes 54 seconds to decimal degrees.
Another example of converting 50 degrees 22 minutes 11 seconds to decimal degrees.
Conversion of negative angles to decimal degrees, such as -5 degrees 48 minutes 41 seconds.
Final summary on converting degree measurements to decimal degrees.
Encouragement to like, subscribe, and hit the bell button for more video tutorials.
Transcripts
[Music]
hello mark awamat in this video lesson
we will discuss angles in a unit circle
so in geometry an angle is defined as
the union of two non-collinear rays
which have the same endpoint
and the definition restricts the angle
measures between 0 degree
and 180 degrees inclusive or exclusive
but
in our discussion in trigonometry we
will discuss in different perspective
view angle
since
angle
will be taught as the amount of rotation
generated when array is rotated about
its endpoint
hence the necessity to label the side of
an angle
as you can see class
of the initial position of the angle
hitting a tower nothing initial side
and then
the position of the ray and the end of
its rotation will be called the terminal
side so the r the arrow head pointed two
terminal side so you arrow head not in
class
a terminal side and then it
takes
no reyna
toward nothing vertex
nothing vertex no anger
the direction of rotation of array also
affects angle measures so as you can see
class
illustration
rotation or another ring and a counter
clockwise okay paganito and
for example
so
say some
rotation
okay a sambal rotation class that is
equal to 360 degrees
since counter-clockwise clockwise
okay so next touches uh initial side not
end and then the end
should say initials initial side ibis
so that is equal to 360 degrees so as
you can see class
again
so that is 180 degrees
and
since dito tayo clockwise your rotation
so young angle nothing is negative so
this is an example of
rotate now clockwise so the angle is
negative 30 degrees
same detail this is negative for 120
degrees
and then
sobake umabu taina 420 degrees
so samaki tangy class
okay similar details initial sky
360
plus
60
so that is 420 degrees so jumbo rotation
yeah 420 degrees so again plus
counter clockwise positive your angle
nothing but clockwise negative yield
angle net
in units of angle measure maritime
delaware the degree measure and the
uh
degree or the regen measure so the
degree measure was developed by the
babylonians around 2000 bc it was based
on sex adjectival numeration system so
unsub is a
sex adjective enumeration system
to one complete rotation of array or to
a revolution is equivalence at 360
degrees so you know assigned and it's
about what degree say some degree is
equal to 60 minutes
and one minute is equal to 60 seconds
so let's try to convert or change
the given angle measure in decimals to
minutes and seconds
so for example maritime
22.4 degrees so nothing shy to change
into minutes and seconds
okay that's 22
and then add nothing in decimal nothing
which is point four so it again degrees
theta is a 20 to nothing plus
zero point four and into zero point four
and next nothing's c degree a minute so
in multiplying that n
0.4 k minutes
so that is
0.4 times
one minute i equivalence ah 60
ah one degree
equivalent to 60 minutes so
convert nothing
so multiply nothing it took by 60
minutes
and then simplify zero point
so 0.4 times 60 that is
24
minutes so copy 22 degree plus 24
minutes so 22 degrees plus 24 minutes
so it opens up
okay so since it's a whole number now so
therefore it's the name final answer
so that is 22 degrees 24 minutes it's a
calculator
capacity
so another example
14.21 degrees so unangled
so 14
degrees plus in decimal point 21 and we
multiply nothing it okay
60 minutes so 0.21
times 60 minutes
the answer is so copy 14 degree plus
0.21 times 60 minutes is 12.6 minutes so
as you can see class melon time uh
i made decimal pattaya so c12 it did not
in illegal minutes so copied 14 degrees
plus 12 minutes and then it on decimal
the point six multiply nathan's seconds
okay so
again class since canina has a whole
number
so therefore in anatomy multiplies the
seconds
so dito
a second so plus
0.6 times 60 seconds
okay 0.6 times 60 seconds is equivalent
to
14 degrees so copy plus 12 minutes copy
then so 0.6 times 60 again that is 36
seconds so ethernet so since whole
number now so
the final answer is
14 degrees 12 minutes 36 seconds
okay so by using calculators
14 degrees 12 minutes 36 seconds so it
14 degrees 12 minutes and 36 seconds
okay
it checked
out
okay next is
to convert the degree measurement to
decimal degree so pano gagavinito so for
example marine time 31 degrees 12
minutes and 54 seconds so you could
convert nothing to decimal degree so una
okay so copy the given and then it says
solution attend start
so in one minute a long second 60
seconds so therefore
say 54 seconds if we multiply now 10k1
okay one minute is equivalent to 60
seconds so
then imma multiply not then
okay palama convert nothing it or some
minutes so one minute over 60 seconds
okay
so once you are getting 54 times 1 over
60
that is 0.9
minutes so mean it's not as amazing
in similar terms insane neces
so that is 31 degrees
12.9 minutes and then it class again
nothing degrees equal convert nothing to
12.9
sa degrees
so 31 degrees plus 12.9
minutes times since again it could
convert nothing it was a degrees
ah unit
1 over 60 minutes okay so paramahamsa
you mean it's my one young degree so
that is 31 degree plus so 12.9 times one
over sixty is zero point two one five so
online class same necessary degrees so
therefore atlanta so thirty one plus
zero point two one five
the answer is thirty one point two one
five degrees okay so again you're not in
sha convert in degree measurements at
decimal degree
next
so another example we have 50 uh 70
degrees 22 minutes in 11 seconds
11 seconds some minutes
okay so 70 degrees 22 minutes plus 11
seconds times one over sixty seconds
and then eleven times one over sixty
that is zero point eighteen so pare
honestly so he had nothing twenty two
minutes plus zero point eighteen
so the answer is
22.18 minutes and then etune among 22.18
echo convert the manhattan into degrees
so that is
70 degrees plus
22.18 minutes times
1 degree over 60 minutes and so 22.18
times 1 over 60
the answer is 0.37 degrees so add 70
plus 0.37
so the answer is
70.37 degrees
next
so it's a calculator padding urine and a
check okay
equals
we have negative 5 degrees 48 minutes 41
seconds
so same process so
41 seconds convert that in some minutes
so that is 1 over 60 41 times 1 over 60
the answer is 0.68 and then 48
plus 0.68 the answer is 4 5 negative 5
degrees 48.68
minutes and then
minutes it will convert nothing into
degrees
so 5 degrees plus
48.68
minutes times 1 over 60 and that is
0.81 degrees so therefore negative 5
plus 0.81
the answer is negative 5.81 degrees i
hope that you know how to convert the
degree measurement to decimal degree
thank you for watching this video i hope
you learned something
don't forget to like subscribe and hit
the bell button put updated ko for more
video tutorial this is your guide in
learning your mod lesson your walmart
channel
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