Standard Position Angles & Radians Part 1

Standard Position Angles & Radians Part 1

BAM!!! Mr. Tarrou We are starting one of my favorite topics which is trigonometry. The study of triangles and their rotations around
a circle. This is used in engineering, measurements of real life items that are too big to do
with a ruler and a straight edge and stuff like that. It has to do a lot with mathematics
of rotation and spheres. We live on a globe. A lot of mechanical items have tons of spinning
items in them, just think of an engine with a spinning crank shaft. So trig is very important
in later studies of mathematics and engineering. Let’s get started with a very basic idea and
some definitions. We are going to introduce standard position angles and try to in probably
a two part video explain what radian measures are…that is another way of measuring rotation
besides degrees. A standard position angle has a vertex on the origin and it’s initial
side on the positive side of the x axis. So if I wanted to start drawing a standard position
angle, I would start off by putting the vertex on the origin and initial side on the positive
x axis. Now the terminal side is the only part that is allowed to move in a standard
position angle. So if you wanted to draw a 45 degree angle…BAM!! This is the terminal
side. Well that is fine but that is an incomplete picture. This is because one of the other
definitions I have to talk about is coterminal angles. Those are angles that have a measurement,
two angles that have measurements that are separated by a full rotation of either 360
degrees or 2pi radians. You might not know what that means yet (radians), but hopefully
by the end of this video you will. Ok, so we have an initial side, a terminal side,
now I need to add an extra part to this diagram to indicate what direction the rotation is
and how much rotation there is. So I am going to draw a little arc that is going to indicate
counter clockwise rotation and i did say that we were drawing a 45 degree angle. Well what
if I wanted to draw a negative 45 degree angle? What would that negative rotation look like?
So I am going to draw another angle on this exact same picture. Well all angles in standard
position have their vertex on the origin, they all share the same initial side, so there
that is the first part of my new angle I am drawing. But, now I want to rotate negative
45 degrees. Well positive rotation is counter clockwise from the positive side of the x
axis, the initial side, and so I think you might guess what negative 45 degrees is…that
is clockwise rotation. Again from the positive side of the x axis, so something like that.
So there we have an initial and terminal side. We have another initial and terminal side
of the negative 45 degree angle. Ok, so there you go! All drawings of standard position
angles have a vertex on the origin, positive side of the x axis, rotate counter clockwise
for positive rotation and clockwise for negative rotation. Now later on in this textbook and
in your class you are probably studying bearing questions. You know how to read a compass?
Compass rotations, bearings, are going to based off of due North so that might be a
little bit confusing for some of you when we get to those questions about bearing. But,
a standard position angle, all of those rotations do start off the positive side of the x axis.
These are not like navigation angles. Now you see I have got I, II, III, and IV up here.
This is just the coordinate plane. This is quadrant 1, this is quadrant ii, quadrant
iii, and quadrant IV. So our 45 degrees is in quadrant I and negative 45 degrees is in
quadrant IV. Now let’s talk about this idea of coterminal angles having angles separated
by full rotations of 360 degrees. I want to draw an angle that is coterminal to 45 degrees.
What is that going to look like? It is going to be in standard position so it is going
to share the same initial side that ALL standard position angles share the same initial side.
ALL of them have their vertex on the origin. So you know this does not say co-initial angle,
it says coterminal angle. Co-habitate, coexist, cooperate…work together or share…they
are sharing the same terminal side. That is the big deal, that is the side that moves.
They ALL share the initial side in standard position. So, an angle that is coterminal
to 45 degrees is going to look like a 45 degree angle but that is not a complete picture.
Is that a 45 degree angle or is that an angle that is coterminal to a 45 degree angle? We
don’t know that without these little arcs to indicate direction of rotation and how
much. I need to rotate and additional full rotation of 360 and I am going to show that
by doing this. Start at the positive side of the x axis, rotate positively because I
want to do…I just decided to do a positive rotation, and I am going to rotate all the
way around until I get back to that same terminal side. Now how many degrees have I graphed
here? I have graphed a full 360 degrees plus an additional 45 degrees to make an angle
that is 360 plus 45 is 405 degrees. Now we could also rotate in the opposite direction
as well. We could instead of adding 360 like I have done, we could subtract 360 and that
will show an angle of rotation instead of going counter clockwise, we would go clockwise.
But if I do any more arcs on this you will not be able to read anything. We could do
45 minus 360 and get a negative, what would that be negative 315 degrees. You can add
or subtract a full rotation and get a coterminal angle. I have one more definition up here
for you. It is a quadrantal angle. I have drawn an angle in quadrant 1 and I have drawn
an angle in quadrant 4 with the negative rotation. If I draw an angle such as let’s say 90 degrees,
if I rotate exactly 90 degrees I am landing the terminal side on an axis line. I am not
actually in quadrant 2 and I am not in quadrant 1, I am exactly in between, this angle is
not in a quadrant it is a quadrantal angle. Any angle with a measurement of 90 or an integer
multiple of 90 is going to be considered a quadrantal angle. It’s terminal side is not
going to be in a quadrant at all. Ok, let’s erase all of this and get on to the next topic
which is how to measure rotation with radians and not degrees. We care about another way
measuring rotation because in certain areas of mathematics, where we are doing mathematics
where things are spinning a lot like engineering, if we are doing navigation around a globe
a round sphere, some mathematics can be done easier with radian measures instead of degrees.
Now let’s talk about a full rotation of 360 degrees. That is not too bad. We have a radius.
So we have a circle, I have rotated 360 degrees. Thus I have got a bottom heavy circle here.
So let’s talk about a measure of rotation that we are comfortable with which is degrees
and see if we can’t convert it into the idea of radians. For radians I want to talk about
this idea of pi. Every time you see the letter pi which stands for approximately 3.14, that
does not mean that automatically you have a radian measure. But pi of 3.14 does have
a lot to do with radians. Do you know where 3.14 comes from? A lot of my students do not.
Well, if you take a circle such as we have here only maybe yours is actually round:)
I am going to have this length of r and I am going to try my best to estimate this length
and wrap it around my not so perfect circle. So maybe somewhere around here and that is
one length of the radius. And so we have one length of that radius. Let’s estimate that
the best we can again and say it is probably somewhere around there and lay down another
length of the radius of this circle. Now we are at 2 radiuses. And then finally let’s
do another length of the radius which is probably about there. Now we are at 3 radiuses. Where
are we on the circle? We are almost to the other side of the circle. So in a measure
of rotation that we are comfortable with which is degrees, I have just about gotten to the
side of the circle…I have just about rotated to 180 degrees. Now I am counting off the
radiuses and I am almost at that 180 degrees. What do you think is left in that little bit
of a gap? What value do you think is in that little piece right there? That little piece
is approximately equal to .14 radiuses. So 3.14 is the number of radiuses, or radii,
to get half way around a circle. Now you can also say that pi 3.14 is the ratio of the
entire circumference with the diameter. 3.14 diameters gets you all the way around the
circle, but all of our formulas are going to be based off the radius and not the diameter.
So, I am going to say, and it is true, that 3.14 is the number of radii to get you half
way around the circle. Half way around the circle is 180 degrees, so 180 degrees is equal
to pi radians. Now where do think they got this idea, this name of radians from? Well
because it is measured and based of the radiuses. So radian is based off of the measure of the
radius of the circle, and so radians is found by doing s over r. S stands for arc length
and dividing that by the length of the radius. So regardless of how big the circle is, there
is only one shape that is a perfect circle, so regardless of the size of that circle 3.14
radiuses is what it takes to get you half way around the circle. Pi radians equals 180
degrees. Nice:D Ok, you also might note that 1 radian gets you somewhere near the end of
quadrant 1. 2 radians gets you into quadrant 2. 3 radians is still going to be in quadrant
2, almost getting you to 180 degrees. I am getting close to running out of time so in
the next lesson we are going to look at converting from degrees to radians and building up all
the angles around what you will know eventually…not in this video…but what you will know eventually
as the unit circle. That is hugely important for you to memorize in the rest of your studies
of Trig. I am Mr. Tarrou. I will be right back:)


  • danradz4me says:

    I needed help on this topic, but when I first searched, your video did not come up and I was scared! Good thing you were in the side for 'related videos'. Just wanted to thank you once again!

  • ProfRobBob says:

    Thanks for not giving up the search…I have added more keywords to the search list after reading your comment, so thanks too for pointing that out:) I appreciate the help.

  • Caden Barton says:

    Great Video!

  • ProfRobBob says:

    Thank You…and thanks for supporting by liking and subscribing!

  • Caden Barton says:

    For sure! Thank you for the great videos. They were life savers in College Algebra and I'm sure they will continue to be all the way through Calculus.

  • ProfRobBob says:

    I always enjoy hearing that! (not that you needed them, just the part that they helped:)
    I hope this means you will be sharing my channel with all your friends and classmates so I can one day have "my dream" fulfilled and I will have a "M" next to my subscriber #!

  • Caden Barton says:

    I always recommend your videos. I'll have to start posting them on my math class's discussion board. Keep it up! Your videos are worthy of that M.

  • videofountain says:

    First impression. Nice retro handwriting. Lost Art. You scared me when you jumped onto the screen like that. That strap on your belt, and perhaps the shirt, is distracting. I like the little metal decoration on the belt. Clear voice. Thanks.

  • ProfRobBob says:

    THANK YOU Caden…My wife and I are looking forward to that day:)…and we'll be one step closer thanks to students like you!

  • ProfRobBob says:

    Thanks for the critique…I hope I wasn't too distracting to learn the lesson from:)

  • AP Economics (and assorted other stuff) says:

    Great lesson. Looking forward to watching the rest and brushing up on my Trig.

  • ProfRobBob says:

    THANKS…always nice to hear feedback from other teachers:) Enjoy your brushing up and beware, you might be saying BAM! yourself before it's over…lol:D

  • ProfRobBob says:

    Thanks:)…and thanks for supporting by liking and subscribing, please spread the word for me too:D

  • BK1_SLC says:

    I feel bad because you make my professor look like a flop. Great video, will be watching these plus my daily lecture. Thank you for the uploads!

  • ProfRobBob says:

    Don't feel bad…feel HAPPY that you found my channel:) I look forward to being your YouTube teacher thru this math course, and THANKS for choosing my channel to learn from! Please share with your classmates if you think they too might benefit and remind them to SUBSCRIBE to show their support and help my channel groW.

  • Deepshikha SINGH says:

    I am very thankful to u for uploading your exceptional vedios ..but is there any way you can upload the vedios related to CSET MATH SUBTEST 2 AND SUBTEST 3 as I am in the process of giving my state tests by December this year.

  • ProfRobBob says:

    Thanks for watching and glad I have been able to help you with your studies. Sorry but I am not familiar with the test videos you are requesting:( All of my current videos are key-worded so maybe you could search for additional help that way.

  • Deepshikha SINGH says:

    Yes, I understand but CSET MATH is a california based teacher examination program. Here is the link for sample questions It would be really great if you can come up with these types of challenging questions made for just teachers examination test. Keep up the good work ..Thanks..

  • ProfRobBob says:

    I will take a look at your link, but I not able to take video request this school year because I am teaching four subjects at once and Calculus for the first time. I am doing everything I can to keep ahead of my students in that class and prepare the rest of necessary videos for Calc.

  • Marien Tactic says:

    i just realized you also have videos on other math subjects like trig and geometry…
    I wish I found your videos last year… 🙁 Would've been a big help

  • ProfRobBob says:

    🙁 me too. But you obviously found my channel page now and know that I have lots of videos still left for you to watch:)

  • Ibadullah Abid says:

    hi sir, this is ibadullah from Afghanistan. your teaching method is outstanding and it's helping me a lot for my entrance test and at last thank u for such a great service.

  • ProfRobBob says:

    Welcome to Tarrou's Chalk Talk Ibadullah and THANK YOU for your support by watching, liking and subscribing! I really enjoy hearing where students are from, I miss that feature! I'm still amazed at how far away I am able to teach from my home in Florida:) Thanks for choosing me to be your YouTube teacher and best of luck and let me know how you do on your entrance test!

  • Ibadullah Abid says:

    well I can't believe that you have responded to my comment!
    there are few months remaining for the test and after I give my test In Sha Allah I will let you know about it

  • ProfRobBob says:

    Just keep watching,learning and preparing…the test will be here before you know it but by then you will be ready to take it with confidence:) Thanks again for watching and I will look forward to hearing your results:)

  • liverbitz91 says:

    Keeping in mind that Pi is not 3.14, it's 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 …(cont'd.)
    However, because this is a calculation to such a small fraction, standard tools used for measurement only go to +/- .005cm (assuming the standard tool for measurement is to the millimetre.) Thus, using Pi to two decimal places is not incorrect, it would just be nice to see a distinction to limit confusion in the future.

  • zyphon7 says:

    I was trying to do trig and having a hard time figuring some stuff out.  Then I watched this video…insta-sub!  lol thanks

  • Fernando Gallardo says:

    yeee excellent video. I shall refer my friends to this page.

  • Mike Doherty says:

    Hello ProfRobBob. I wanted to say thank you for this, and the other, videos. It's been a loooooong time since I undertook formal studies but I want to build competence in maths so I feel confident looking at some areas of physics that really interest me – quantum mechanics, thermodynamics et al. Just for the fun of it.

    Very early stages, of course, and a long way to go but I find your enthusiasm, energy and general approach makes for easy learning. So here's to the next 200 hours of study. 🙂

    Just off to 'like' the other vids I've watched ( and Facebook too).

    All the best.

  • AwkwardPasta says:

    my professor isn't really good at explaining things. so I've been taking notes from your videos and i understand it better! You're pretty much my math teacher now! Haha

  • SuperGrey95 says:

    Thank you I really hate these lessons but your explanation make them better for me, thank you Prof 🙂

  • Rhi Elle says:

    Hello Mr. Teriou, I apologize if i spelled your name wrong. I wanted to thank you so very much for uploading these math videos. I've been watching your videos for the past couple units i've had and you've certainly helped me to understand the concepts so much better. Your explanations are clear and precise and make sense, thanks for breaking it down and explaining so thoroughly. My only wish is that you were my Prof. God bless you…

  • Reeman Tineh says:

    Hello Mr. Tarrou I have one question. What book do you you use in you classroom? Thank you very much. 😀

  • xxfaction6xx says:

    best vids man. thanks prof

  • Jonas Gilje says:

    Thank you for making these videos!
    I just have a little question about trig.
    Why is tan(theta)=sin(theta)/cos(theta)?

    Thanks from Norway!

  • Jordan Earl says:

    these videos are really helping in my pre med classes where i'm trying to brush up on everything. THANKS SO MUCH

  • Alexander Ablaza says:

    You are very awesome great fantastic incredible superb teacher sir. I will be your student for the rest of my life.

  • Jasmeen Kaur says:

    hey mr. tarrou how do I find the six trig functions for an angle less than 0 like -120 degrees?

  • main says:

    Mr. Tarrou, would a precalculus book be a good replacement for a trigonometry book? And vise-versa? Any book recommendations? Love your videos; you really make math interesting. I'll be sure to like every video I watch.

  • saad merhi says:

    thank you for helping me passing and getting a good grade in my pre-calc class, you a truly an awesome prof. 

  • Syed R. says:

    love it

  • James Strzelecki says:

    Pokeball! Hahahaha!

  • Cory Baker says:

    Ugh, all I can think whenever I'm doing anything trigonometry-related is how much easier it is when using Tau.

  • Honer Zawita says:

    I've got what I want from Radian measures, I'm happy because now I understand it. Many thanks.

  • Kiyoko Kamiyama says:

    Hi Prof Rob! Do you have a video about converting degrees to radians or vice versa WITHOUT using a calculator? 

  • Colette Klassen says:

    I used to struggle understanding the connection between radians and degrees and was frustrated most of the time now that I know the history behind radians and how they cam about it all makes sense my fear just dissipated. I hope to be as thorough and day when I teach my students here in South Africa. I am becoming more and more excited with every video I watch.

  • NoA Naga says:

    My question is, if you have to find two co-terminal angle of -15.5(degree), but you are restricted on your co-terminal angles. In another words, your answer has to be greater than zero(degrees) but less than or equal to 360(degree). I found one co-terminal angle which is 344.5 (degree), but i am unable to find the other positive co-terminal angle.

  • Scieneering says:

    Can't wait to watch all your trig videos for this semester! Do you teach at a high school? If so what grade do you teach and what math do you teach? Or do you do all different maths at a college? Thanks for all your help BAM!!

  • Ishhy Navarro says:

    I have exams tomorrow on this and you made everything clear! I learned more from you than the whole quarter I'm learning in school. THANK YOU. YOU ARE A GIFT.

  • Pure Awesomeness says:

    Hahaha. I love the Bam!

  • redcentredoggies says:


  • Dovid Baum says:

    you are my lord and savior. thank you so much

  • Your Friend Primm says:

    Man, all your videos are awesome! I'm currently in the Navy and my options are limited and I have to take my math classes online…no instructor. I am currently taking trig and I was struggling at first trying to teach myself from the book, then I found your videos. I am very grateful for them, and I am thankful of the way you break the information down and show all the work in a way that really clicks for me me and I would recommend your video's to anyone! You are a great teacher!

  • Ayush Mishra says:

    Do you have trigonometry for Algebra 2 instead of Precalc?


    superb ☺

  • asaph vega says:

    hey mr toru?? I just want to say this video is amazing and very helpful however I can't get this feeling out about that ID Lace of yours that dictates "devil"??? I just want a clarification if you don't mind?? no offense sir

  • stephonsjx says:

    lol pretty sure the plural of radius is radii

  • Alicia Sierra says:

    this was so aesthetically pleasing & very helpful thank you 🙂

  • 迟翔 says:

    hi, sir my country blok youtube and I only could watch it in school is there any other way I could watch your video ?

  • Nadia biglari says:

    Professor. Tarrou , i got a B in trig, thank you. next semester taking pre-calculus, expect you to be there for me.

  • Craig Coates says:

    you literally are amazing, before this video i hated radians and never saw the annoying point of them or there origin, but this is brilliant and clear than you

  • SupremeBeing says:

    Correct if i am wrong but wouldn't angle negative 45 degree be located in the fourth quadrant.

  • Shalewa Thompson says:

    do you cover all the topics in geometry  in your videos

  • Aldous Torres says:


  • Mr. Lonely says:

    hope this will help me, cause I'm failing trigonometry on 2nd semester. i got failing grades. its because i dont really get this topic.

  • Homer Noble says:

    Horray! Mr.Tarrou can u make a video about right spherical triangle and Oblique spherical triangle?? I cant find any relevant video tutorial on YouTube and we're having our Finals nxt week. I hope u can read this and help us by making tutorial! more power in this channel and god bless 😀

  • Letoman Let it be says:

    now I know where the idea of pie radian came from, thanks for the wonderful video, Bam!!!!

  • tenton2000k says:

    At 3:48, Rob refers to "this textbook." Anyone know and have the link to what textbook he is using?

  • YJ Jung says:

    Is there any textbook that I can study with this video? Do you have any recommendations?

  • Phil Galatioto says:

    Thank you professor :)))

  • RevolutionaryComics says:

    you are amazing. nuff said.

  • zane Smith says:

    studying for the FE thanks for the vids

  • meme says:

    Do you teach school? This is awesome!

  • adam love says:

    robert uncle terry said you did a great job

  • adam love says:

    Robert we really appreciate talking to you thank you for being here.

  • raven says:

    You sir, are da real mvp! I have a useless pre-calculus teacher who's always not around and just leaves seatworks and we're having a test about this tomorrow. I'm so glad I found your channel because you explain things perfectly and I don't get sleepy with your teachings. After I binge watch your other trigonometry videos maybe I'll actually pass the exam. Wish me luck 😂

  • Amil Jayasinghe says:

    is it me or did he just said freakonometry ?, he is one of the best teachers I came a cross though

  • Norah Al-Mogrin says:

    You saved me from my struggle with pre-calculus…..
    THANK U SO MUCH with your useful videos i think i'm gonna pass my final exam

  • Sandra Pal says:

    Thank you so much! I wish I could replace my current pre calc teacher with you – maybe then, i'd actually ace my tests

  • F2F Saberi says:

    I have a lot problem with math😢😣😭help me please

  • Ionic-Android says:

    The comedian.

  • Vu N. says:

    Thank you so much 🙂 !

  • Sidney Goodrich says:

    When you explained where radians come from and how they are related to angles, I was just totally mindblown. I literally just paused the video and smiled for a few minutes. You are made to be a teacher. Thank you so much. Keep doing what you're doing! Already subbed 🙂

  • Ej_ L says:

    what a nice introduction to trigonometry! Looking forward for more videos like these 😀

  • may 24 says:

    thanks so much this is very clear and easy to understand

  • soviethammer says:

    Does anyone know of a good website to find practice problems so I can cement my knowledge?

  • jaison antony says:


  • jaison antony says:

    everything is awesome……..everything is awesome……………

  • Mohan7 says:

    Whenever your video starts, I wait with missing heart beat for your jump into the screen ! That is an awesome start, wakes up sleepy students like me . Lol. Thank you for all your videos.

  • Salvatore G says:

    This man is a genius. Thanks so much for the great videos!

  • Mohan7 says:

    I tipped you a dollar today. I wish I could give more. That's all I could afford. I felt guilty for watching all these greatest math videos for free !!

  • Jerry Webster says:

    Hello. Thank you for all of your excellent videos and the "Bam!". Would have loved to have had some crackle back in High School, let alone some bam. Alas, it's never to late to relearn long forgotten concepts. Which book do you currently teach trig with?

  • Moon Stars says:

    how is the negative 45 angle In quadrant 2 !!!!!!!!!! (4:25 time )

  • Michael Jennings says:

    Your video lectures are awesome but I think you are missing something. It would be even better if you added a video or maybe two on co-terminal angles.

  • maximilian2000 says:

    Why does he keep on saying radiuses? It's a Latin word, and the plural is radii.

  • livid says:

    8:00 umm hes just levitating?

  • CO -POWER says:


  • Eschatonx says:

    Thank you for uploading your lesson. I'm 36 years old and it has been almost 20 years since I last took a math class. I only ever got to statistics on my in my last degree. Coming back to school to get a math degree and conquer one of those fears I've had in school my entire life. Your videos let me find out what I need to refresh myself on from algebra before taking on these classes in the future.

  • Thomas Barber says:

    Mr. T, you are my favorite YT maths instructor; so, please try not to SCREEECH the chalk.

  • Dane Calderon says:

    40 years old and I just now learned where 3.14 comes from. Hopefully my son will ask me about this some day so I can look smart!

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