Law of Cosines - Formula, Examples & Proof - [2-20-4]

View more at www.MathAndScience.com. In this lesson, you will learn about the law of cosines in trigonometry. We will focus on presenting the formula for the law of cosines, then we will work several examples to get practice, and we finish the lesson by proving the law of cosines. We can use the law of cosines for any triangle, including triangles that are arbitrary in shape and not right triangles. We can think of the law of cosines as being a modified Pythagorean theorem c^2 = a^2 + b^2 where we add an extra term to correct for it being a non-right triangle.

Пікірлер: 69

  • @kate6370
    @kate6370 Жыл бұрын

    Sir, I became your subscriber last 2019! I'm so happy that you're still active here on KZread. Thank you for sharing your knowledge to us!

  • @sohampatel6122
    @sohampatel6122 Жыл бұрын

    Hello Jason! You are a very great teacher. For me, trigonometry was very hard until watching your videos step by step which made me very proficient in the subject! I even started explaining your subjects to my family and they became proud! Keep up the great work, thank you.

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    You are very welcome!

  • @monak4854
    @monak4854 Жыл бұрын

    The gift that keeps on giving. Thank you 🙏

  • @ullastvtl
    @ullastvtl9 ай бұрын

    Trigonometry was my dreaded Bugaboo during my teenage. Today at 61 years, I understand it without fear. Thank you very much, Mr. Polymath Jason.

  • @namazovmath3372
    @namazovmath3372 Жыл бұрын

    ❤I was waiting for this.Thank you very much

  • @Carlos-qz7ul
    @Carlos-qz7ul Жыл бұрын

    Brilliant! ❤ Mr. Jason has me levitating as a feather over a canyon...

  • @simpleman283

    @simpleman283

    Жыл бұрын

    Me too, no doubt. I worked 5 days, on understanding the proof at the end. Today, it clicked & i've got that same feeling.

  • @efremtse6518
    @efremtse6518 Жыл бұрын

    Thank you so much you teaching me a lot.

  • @miguelvenegas4172
    @miguelvenegas41727 ай бұрын

    Absolutely amazing teacher!! Wow.

  • @vascobishop3359
    @vascobishop33596 ай бұрын

    Good teacher who pick some excellent points and topics. Has anyone seen his unit of circle lesson? Amazing content.

  • @simpleman283

    @simpleman283

    3 ай бұрын

    Many times. So many times in fact that I noticed a pattern on it just the other day, that now I can even figure what the radian of any angle on the unit circle will be. You are correct, Amazing content, indeed.

  • @animeislife6428
    @animeislife6428 Жыл бұрын

    you are the greatest teacher for us

  • @junmaclan26
    @junmaclan26 Жыл бұрын

    Wow! Thank you so much Jason.

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    Welcome!

  • @oo-pi2dq
    @oo-pi2dq Жыл бұрын

    Thanks ♥︎

  • @racksvere4726
    @racksvere4726 Жыл бұрын

    thanks sir 🙏🙏

  • @molirathod1407
    @molirathod1407 Жыл бұрын

    Nice teaching sir ! 👏👏👌👌

  • @Guran998
    @Guran998Ай бұрын

    He makes it so much easier 😮😮

  • @tresajessygeorge210
    @tresajessygeorge210 Жыл бұрын

    THANK YOU... SIR...!!!

  • @keabetswekwenda5760
    @keabetswekwenda5760 Жыл бұрын

    Thank you very much sir it was helpful I used to struggle a lot before I started watching your videos you are much appreciated

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    You are very welcome!

  • @cicerochen313
    @cicerochen3139 ай бұрын

    awesome!!

  • @liasitumorang6041
    @liasitumorang6041 Жыл бұрын

    Thank you sir, GBU

  • @drumtwo4seven
    @drumtwo4seven Жыл бұрын

    Nice 👍

  • @vinuVA
    @vinuVA Жыл бұрын

    We don't all have to be students to learn, all we need is interest and willingness to listen.

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    Great advice. Love this

  • @WahranRai
    @WahranRai Жыл бұрын

    the theorem of Pythagoras is for the right triangle, for other triangles the law of cosines or *Al Kashi* 's theorem

  • @donsena2013
    @donsena2013 Жыл бұрын

    My preference is first to derive the law of cosines from the Pythagorean Theorem and to show the Pythagorean Theorem to be simply a special case of the law of cosines, as the included angle in the cross-product term approaches 90 degrees, thus zeroing out that last term. The law of cosines can be derived from a simple vector analysis, starting with the definition of the dot product, and independently of the Pythagorean Theorem.

  • @irishamerican4558

    @irishamerican4558

    Жыл бұрын

    I doubt he cares what your preferences are.

  • @donsena2013

    @donsena2013

    Жыл бұрын

    @@irishamerican4558 He should. Law of cosines by use of the dot product: kzread.info/dash/bejne/nnWk0pdwe9Wrdag.html

  • @bymoviestube9202
    @bymoviestube9202 Жыл бұрын

    Ur the best explaining teacher both for math and physics. I've subscribed scince I was elementary and your slow step by step way of explanation helped me experience the early world of math and science without stress or fear. I really mean it when I say 'thank you'.

  • @HeavenlyPress
    @HeavenlyPress Жыл бұрын

    17:38- that entire tools section was immaculate

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    Thank you so much I appreciate it!

  • @niyonkuruidrissa2346
    @niyonkuruidrissa2346 Жыл бұрын

    10x for your commitment sir!

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    Thank you!

  • @eipimath
    @eipimath Жыл бұрын

    Thanks for sharing! I posted a short video on deriving the law of cosines, It should apply for all angles (acute, obtuse, reflex, negative). Hope to get your thoughts.

  • @simpleman283
    @simpleman283 Жыл бұрын

    9:40 first problem. 16:23 negative times a negative = positive. My mind tells me this really is subtracting a negative 689.4, which makes it positive. 30:50 31:55 "make sure you understand this before we go on any farther" Well it took me 5 days, but I fully understand it now.

  • @and4563able
    @and4563able Жыл бұрын

    When will these lessons be posted on the website?

  • @adotty1186
    @adotty1186 Жыл бұрын

    I love you mate

  • @simpleman283
    @simpleman283 Жыл бұрын

    This is good, I've been working on law of Cosine & Law of Sine. Me & my calculator are not the best of friends just yet. I got the first one right, but it took me 7 minutes.

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    Remember fast isn’t the best. I’m not fast either.

  • @davidwarui2073

    @davidwarui2073

    Жыл бұрын

    Lovely,just lovely listening to you teach calculus and marval at understanding what I didn't while attempting chemistry in 78

  • @gustavogaviria7678
    @gustavogaviria7678 Жыл бұрын

    Hello Jason, please let me know how I can contact you as I have some questions regarding the paid membership.

  • @bobaprakash8905
    @bobaprakash8905 Жыл бұрын

    You are awesome

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    Thank you!

  • @simpleman283
    @simpleman283 Жыл бұрын

    31:55 "make sure you understand this before we go on any farther" Well it took me 5 days, but I fully understand it now.

  • @4everThoughtful

    @4everThoughtful

    Жыл бұрын

    That's the spirit!! There is no learning math without a struggle. If you're not struggling with it, you're not learning it; or you already know it. Awesomeness!

  • @simpleman283

    @simpleman283

    Жыл бұрын

    @@4everThoughtful Thank you so much for the comment. After working on that problem a couple days on paper, I tried working it on my mid sized whiteboard. Frustrated at not being able to solve it, I placed it where I could see it even while watching videos. I would look at it from time to time. Then I noticed another way to try. Just a couple times doing that & I had it. I really feel like the whiteboard constantly being visible and not being in a classroom rush was key. It was such a feeling of victory, I had to make a comment.

  • @4everThoughtful

    @4everThoughtful

    Жыл бұрын

    @@simpleman283 I'm sincere in my joy for you. Math is indeed a language and it DOES take persistence. For those that never got it, it's because they gave up; didn't want to struggle. The brain is an awesome muscle and it needs exercise to give you the results that you want. I, too, love a large board that I can write on in large print & see it often.

  • @freepilot7732
    @freepilot77325 ай бұрын

    It seems to work for me. I only remember the last one. I just retitle the triangle.

  • @treesajoseph9255
    @treesajoseph9255 Жыл бұрын

    🥰🌹🥰🥰🥰

  • @bidarvandi
    @bidarvandi Жыл бұрын

    you are a light in Darkness(ignorance,blindness)

  • @vlad_makes_stuff
    @vlad_makes_stuff Жыл бұрын

    Why are the smart channels like this one less popular than the not-so-intelligent Mr.beast and Skeppy? Reply down below!

  • @simpleman283

    @simpleman283

    Жыл бұрын

    Because channels like this are for people that seek to better themself. People flock to those channels you mentioned because they have no plan for their life, they just want to be entertained.

  • @shaunalem4810

    @shaunalem4810

    Жыл бұрын

    Normally people do not like to think critically, so more of entertainment 🙄 world. But this kind of channel requires critical thinking, at least basic thinking to comprehend at least for minutes. So they don't prefer this much.

  • @brucea9871
    @brucea9871 Жыл бұрын

    Good video for those unfamiliar with the law of cosines (although I learned it a long time ago; in fact I taught myself trigonometry from textbooks long before I learned it in school). I do have a few suggestions though. First it's a bit distracting (for you and your viewers) for you to continuously read your notes before copying them to the whiteboard. I could write virtually everything in this video without referring to handwritten notes (the exceptions would be things like finding cos 20° and multiplying it by 2ab). For things like that maybe you could have notes posted just off screen. Second you do not really need three separate formulas for the law of cosines. Instead of having separate formulas for a squared, b squared and c squared we could just relabel the sides of the triangle (e.g we could relabel side a as b, side b as c, and side c as a). Or we could use the general rule you mentioned. The square of any side of a general triangle is equal to the sum of the squares of the other two sides minus twice the product of those two sides and the cosine of the angle between them. Third you could have mentioned the law of cosines can be used to determine the angles of a triangle if the three sides are known. All you have to do is solve the law of cosines for the cosine of the angle and then use the inverse (or arc) cosine button on a calculator to obtain the angle. But perhaps you omitted that to avoid making the video too long. I am glad you included a proof at the end. Although I've seen the proof before others may not have and I think it's a good idea to know where such formulas come from and not just blindly believe them.

  • @simpleman283

    @simpleman283

    Жыл бұрын

    Critique your own video, Oh your channel doesn't have any content. How very interesting of you Professor.

  • @MathAndScience

    @MathAndScience

    Жыл бұрын

    Any university professor will have notes they lecture from. I’m not blindly copying them. It’s to ensure that I say and talk about every topic. If I didn’t have notes I would inevitably miss something. I doubt you could memorize a 30 min lecture. Believe me I have tried that. Again, go to any college class and you will see professors with either slides or notes they lecture from. Thanks, jason

  • @brucea9871

    @brucea9871

    Жыл бұрын

    @@simpleman283 That's as ridiculous as me telling you to critique your own comment. I will critique any video that merits criticism. Essentially you're saying we should not criticize anyone but ourselves which is ludicrous. I am not endorsing criticism just for the sake of criticism (I've seen far too many people do that) but rather constructive criticism so the recipient of the criticism can improve.

  • @simpleman283

    @simpleman283

    Жыл бұрын

    @@brucea9871 No, you are ridiculous. You have NO constructive criticism so, leave the teaching method to the one that doesn't need your help.

  • @simpleman283

    @simpleman283

    Жыл бұрын

    As soon as I read you telling him to change the sides of the triangle, rather than show three formulas for the Pythagorean Equation, I knew you do not understand the importance of students recognising patterns in math. Therefore my comment, Bruce.

  • @user-mg4uz6ix6w
    @user-mg4uz6ix6w2 ай бұрын

    More of this "right?" too many times. Ends up being annoying as far as presentaion skills. Perhaps knows the topic but don't know how to present it well. "stick with me" means I know I over-explain and you guys already got it but I keep saying it anyway.