Advanced Functions 7.3 Double Angle Formulas
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- เผยแพร่เมื่อ 30 ม.ค. 2025
- In this lesson we will derive the double angle formulas, which is really easy now that you already know the addition subtraction formulas. I go through several example of what your homework assignment will focus on. Here's the link for some homework problems that I took up with my students. # 3a,c, 4, 9, 10, 13c, 15c mshavrot.pbwork...
In the last example there should be a radical sign over 2 - sqrt2 in the numerator ... (see, even teachers make silly mistakes!).
My teacher explained 7.2 perfectly but I got lost during 7.3. Glad to know I can always come back here to learn anything I might’ve missed in class. You really are helping a lot of people so I appreciate that. Especially since it’s a Canadian-style teaching, almost identical to the way my teacher does it (except for the fact that teachers have to stretch out a lesson to be REALLY long so we don’t sit around doing nothing for an hour, while these videos are very straightforward). Thank you again, hope TH-cam pays well. 🤑
You're most welcome. TH-cam doesn't pay enough to pay for all the paper I've used. LOL
Thank you so much! I really enjoy the chapter reviews at the end. They make me see what I still need to work on. But these individual lessons during this time is way better than what I learn online. Thank you again!
That’s so nice of you to say! Comments like yours make me SO happy! Hope you come back for Calculus and Vectors and bring some friends! 😊
ms havrot you’ve have been my savour this unit, my teacher has the TA teaching it and i’ve nvr been so lost in class, these videos will be the sole reason i pass this unit, thanks a million for ur super clear explanations ❤️
Probably the worst chapter to pass on to a TA! I think it’s true most difficult chapter of the course. Glad I could help.
Hi Ms Havrot. Thank you so much for all of your videos. I have watched almost all of them from grade 11 functions to now, almost finishing grade 12 advanced functions and they have been so helpful. You explain everything in a way that makes so much sense and I really appreciate these videos. Again thank you.
So nice of you to comment ❤️ it’s students like you who take the time to thank me that make me feel so proud of the work that I put into my channel. Thank YOU for watching and I wish you all the best in your future studies.
Ma'am, shouldnt there a square root on the numerator, at 21:49?
Yes, you are correct!
That's exactly what I wanted to ask .
Oh my god you are the most helpful math you tuber out there! I think you saved my grade!
That’s great to hear! Thanks for the lovely compliment 😊 Please like, share and subscribe to increase the channel’s visibility ❤️
Ms. Harvot, you a real one fr for this - our teacher left us with just the filled in note but i understand it now lol
Always here for a second explanation 😊. Thanks for watching.
Thank you so much miss ! I love your videos, its really helping me get through online advanced functions
I’m so happy that I am able to help you through this nasty time! 😊
i’ve had grade 10, 11, and 12 trig online because of covid and videos like these are really helpful
I'm so happy that the videos are helpful for you. I try to make things as simple as possible and take a lot of time to explain things in an easy to understand way. I have taught so many students over the years that have come back from university to thank me that I thought it would be great to help many others. Thanks for your comment : )
Your videos are super helpful I’m just so relaxed after I’ve visited your TH-cam ( I was very tensed bcuz I wasn’t understanding a thing!) but now I’m good. All Thanks to you!! Hopefully I’ll pass my exams with some excellent results 🤞
So glad that you find the lessons helpful! All the best on your exam ❤️ Please subscribe, like and share to improve the channel’s visibility. Don’t forget that the entire calculus and vectors course is here as well 😊
the haters that left the 2 dislikes do not deserve to watch Ms Havrots INCREDIBLE videos
Ah, there are always haters out there! (probably someone who has a math channel, I would guess!) Not my style to dislike videos knowing how much work people put into their videos. As Thumper said in Bambi ... "If you can't say something nice, don't say anything at all" ... haha
Just dropping a comment to support this amazing channel!!!!
Thanks for the support! Glad that the lessons are helping you prepare for university.
Very good video! For the last question, in my point of view, there shouldn't be +/- in the front since sin(pi/8) is obviously a positive number. I'm I right? 😁
Yes, you are correct. There is only the positive answer as you are in the first quadrant.
Thanks for your comment.
My teacher explained double angle formulas in a very complicated way, thank you so god damn much for this video, this was all I was struggling on for the unit.
That made me laugh 😂. “The essence of mathematics is not to make simple things complicated but to make complicated things simple.” (S Gudder)
Hello, I love your videos! I have shared it with my friends and even to my dentist (because he mentioned that he has a son in grade 11 and is doing functions)! Quick questions, I understand how to do these questions but I don’t understand the what the answers mean. 10:21 Is -7/25 a ratio?
7.2, I understand the meaning behind the answers because we are solving for compound angles and that is useful. but I don’t understand the purpose behind knowing cos2x. I hope my question makes sense! Thank you for your help! 😊
Thanks for sharing my channel! Yes, the answer is a ratio ! cos2x allows you to find exact ratios for double the size of an angle. You will also use double angles in calculus and learn how to take their derivatives. (See you in calculus and vectors?)
ohh okay, I think I understand it (if i have any more questions later on, I ask you) and yes! I'll see you in calculus and vectors next semester and I am going to continue math in university (because I want to become a math teacher too!) so I will be watching your "Beyond High School Calculus" videos as well!@@mshavrotscanadianuniversit6234
Thank you so much for the Videos!!! Was feeling incredibly overwhelmed and Nervous when we did this in Class today as It can get incredibly confusing online but this helped me some of the key concepts I missed out on
That’s great to hear! I’m here for you! ❤️
Your lessons are beyond helpful, thank you!!
I'm so happy that I am helping you with your studies. "Math is hard" (said Barbie) but it isn't when you have a good instructor and you apply yourself, right?
18:22 how did cos2x become 1/√2? i know that for cos π/4, x is π/4, so why isn't it cos2(1/√2)?
Sorry for taking so long to answer. I’m on a vacation. Basically we are letting 2theta be pi/4 which is 1/sqrt2
ms havrot is the greatest of all time
Thank you ❤️. I’ve done my best to make math as easy as possible 😊
Thank you so much for the videos :) they are super helpful and I really appreciate you taking time out of your day to make and post these!
Thanks for your lovely comment. I really appreciate YOU taking the time to comment. :) Please like, share and subscribe to increase the channel's visibility.
For question 1, where you drew the triangle that had y=3 and r=5 how did you come up with -4 as the bottom length?. I understand the negative sign but I actually dont understand how you got the number 4 itself.
Pythagorean theorem. It’s a 3-4-5 triangle 😊
Thank you so much for the math videos! they are really helpful and have benefited me in class.
Your welcome! I’d be really happy to know how much you think these videos have improved your mark. 😊
Hi Miss! How do you know which double angle formulas to use when you have to solve for a single question. One of the examples my teacher went over was "Evaluate sin pi/8" how do you know if you should use one of the cos angle formuals with sin or just sin2x ?
Tricky lesson, isn’t it? You have to think of which formula to use that would best suit each individual question. You might choose one that isn’t going to work and you will figure that out pretty quickly as you work the solution. cos 2x = 1 - 2(sin^2) x works for sin pi/8 because you should be able to see (with practice) that if you double the angle you will get pi/4 but still maintain sin pi/8 on the rhs and with some simple manipulations of the equation, come up with just sin pi/8. There really isn’t a magic answer to your question but rather practice that will help you identify which double angle formula to use.
Hi miss! amazing video, it was really helpful. For the last example, why wasn't the numerator (2 - root 2) rooted while 4 was??
Because I made a mistake! There should be a radical sign over the numerator as well. Unfortunately with TH-cam you can't even add a card any longer notifying of a change. I did add it to the description as someone had asked this before as well (all the smart kids! ) : )
Glad you found it helpful!
Ms. Havrot, im having trouble proving the fpllowing trig identity 1 + cosx ÷ sinx = cotx/2 the most confusing part for me is the x/2. I know that you need to use double angle formulas to solve but should i multiple both sides by the 2 first or is that something you cant do?
Is it (cotx)/2 or cot(x/2)? With the first you can multiply by 2 but the second is the angle/2 so no.
Also is the first part (1 + cosx)/sinx?
@@mshavrotscanadianuniversit6234 yes its [ (1 + cosx)/sinx ] = cotx/2. The cotx/2 is just written as cot (x/2) but without the brackets around it
The 2 is under the x only though
Sorry for the late response but I had several family issues to deal with.
You need to use two double angle formulas knowing that the rhs involves half angles so you can use sinx = 2sin(x/2)cos(x/2) which comes from sin2x = 2sinxcosx
And in the numerator
1 + cosx = 2 cos^2(x/2)
Then you will see that it is easy to simplify to cot (x/2)
Hope that helps 😊
miss your videos are extremely helpful! thank you for all your time and effort! I have a question tho.. in Q1 it said to find sin 2 theta and cos 2 theta. why did you only find sin 2 theta and not find cos 2 theta? is that how we are supposed to solve the question? if we do find the cos 2 theta, what would the answer be?
Yes! I forgot to do it, didn’t I?!! You can do cos 2 theta just like in the very next question except you will substitute in (-4/5) for cos theta. So that will give you cos 2 theta = 2(16/25) - 1. Or 7/25. 😊
@@mshavrotscanadianuniversit6234 yess! I did that thank you! also, do you teach in a high school in mississauga or toronto?
No, I have always taught in Ottawa. Good luck with your studies although luck will have little to do with your success! 😊
@@mshavrotscanadianuniversit6234 unfortunately, you’re right but thank you so much miss! Once again, your videos truly are life savers!
I can't thank you enough.
You just did 😊❤️
Best class ever!!
Thanks so much! ❤️
Hi Ms, I had a question about the last question. So the identity says 2 theta, how come we replace the cos2x with just pi/4 then?
Because in this case 2theta is pi/4 ... perhaps when I said theta was pi/4 it was confusing to you.
I should have said that 2theta was pi/4 which makes theta pi/8
Thank you so much!! your explanation is really good and your lessons are so helpful!
You are most welcome! Thanks for watching ❤️
Thank you so much for your videos, they are very helpful and really clear up any questions I have!! On behalf of everyone watching your videos I can say you really are a lifesaver!! Thanks again Ms. Havrot!!
What a wonderful message ❤️ it makes me very happy to know I’m making a difference and helping students achieve their goals. Thanks for watching! I’m eagerly waiting to see my 15,000 subscribers, hopefully for an early Christmas present 🎁
these videos are incredible!!
I’m glad you think so! I’m not sure my students in the classroom appreciated me as much as my online students have! 😆
Can you take up Question 8 on the Nelson Textbook MHF4U 7.1 pg 407
If you can like a paper answer please, if not typed will work. I spent 1 hour figuring how to solve!
check this solution manualchrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/msmamath.weebly.com/uploads/6/4/8/5/64857803/mhf4u_chapter_7_nelson_solutions_manual.pdf
How do I follow or navigate to your previous lesson playlist?
When you go to youtube and find this lesson simply click on the picture of me and it should take you to the channel with all of the playlists. Let me know if this works for you. : ) Thanks for watching!
@@mshavrotscanadianuniversit6234 yes it worked... Thanks
Looking forward for more topics
hi miss! what did you do at 7:41 to get the x value of 4?
It’s just a right triangle with hypotenuse 5, and a side length 3. So you can use Pythagorean theorem or you should recognize a 3,4,5 triangle.
@mshavrotscanadianuniversit6234 Ohh!! My apologies for the silly question. Thank you so so much for what you do :) I am not exactly proficient at math but you have helped me so much these past few years
No silly questions. Only silly people who don’t ask questions when they are confused.
Very helpful 👍🙂
and for the last example how come you didn't multiply cos with 2? thank you for the lesson btw very helpful!!
For Cos(2theta) we are replacing 2theta by Pi/4 so that we can solve for sin(Pi/8)
I did, however notice that I forgot a radical sign in the numerator of that last question. Let me know if this helps.
@@mshavrotscanadianuniversit6234 thank you!
So easy to follow thank you so much
Amazing video, Amazing explanation
Why thank you! So glad that you have enjoyed this video. I do my best to make everything as clear as possible.
for number 4 why does it become sin x
It’s a cofunctiion identity
Thanks so much Ms Havrot :))
You are most welcome!
ur the best!
Thank you! Just trying to help make math easier for you! 😊
ms havrot you are so réal love you lots!!! xoxo 👍🏻
Well, thank you ❤️
why does cos^2 x + sin^2 x =1
Go to 3:27 on this video for a complete explanation of the Pythagorean relationship.
th-cam.com/video/GCEBhegMMM0/w-d-xo.html
Another way of thinking about it is that when you are on the unit circle, the coordinates of any point on this circle are (cosx, sinx) ... this is because cosx is adj/hyp or x/r and r is 1 so x = cosx ... the same could be concluded for the y coordinate. Now that you know that you can use Pythagorean Theorem (x^2 + y^2 = r^2) and that will get you to the same conclusion. Does that help?
Great thank you
Thank you so much!
Thanks for the help
Amazing!
Hello Ms. Havrot, could you please help me solve this question:
1. Create an equivalent expression for:
cos2x -(sin2x/sinx)
Your only choice is to write sin2x as 2 sinxcosx which would then allow you to divide out the sinx in the denominator , then you could change the cos2x to 2 cos^2 x - 1 leaving you with : 2 cos^2 x - 2 cosx -1
You can check if they are equivalent by substituting in a value for x. I chose 30 degrees (and made sure my calculator was in degrees) which gave me an answer of - 1.232 for both proving that they are equivalent.
No clue what you did on pg 407 q9
Not sure if you read the comments below but there is a small error in the last step where the numerator should still be under the radical sign.
LAST question... in the last step.... you forgot to take the square root of the numerator .
Yes, you are correct. Another student pointed that out to me as well. Always check the comments for a video for "updates" but it's not possible to easily correct a small error without redoing the entire video : ( BUT ... what that does mean is that you have it figured out!! ( When I would make small errors in class I would give the student that noticed it a bonus point on their unit test ... sorry I can't do that for you as well! )
very helpful
Glad to hear that! Please encourage your friends to subscribe : ) So many people don't bother to click the subscribe button and it's the best way to support the channel.
I wish I'd found your channel earlier
Ah, but you are here now. Full speed ahead! : )
nice
Thanks