Trigonometry Proofs Involving Half and Double Angles

Trigonometry Proofs Involving Half and Double Angles


BAM! Mr. Tarrou. Welcome to my third video
about dealing with half angle and double angle identities. My goal in this video is to start
working on some and show how half angle and double angle identities they work inside some
proofs or verify some identities. I am going to do four examples and hopefully get done
in the time span of this video. So let’s get started. The sine of two theta is equal to
two cotangent of theta over one plus cotangent squared of theta. Ok. Well, the sine of two
theta you might know, you should know if you have your memorized and if not start working
on it or at least look at your formula sheet, is two times sine theta cosine theta. So I
need to have all of this written in terms of sine and cosine and do some algebra and
see what happens to get the two sides to match. Well, what to do…what to do. The first thing
to do is go BAM! this is cosine over sine and this is cosine over sine, find common
denominators and do all that good stuff, and it is going to look something like this. Two
cosine theta over sine theta over one plus cosine squared theta over sine squared theta.
We need a common denominator so we are going to take one and make it sine squared theta
over sine squared theta. That is going to become two cosine theta over sine theta all
over sine squared theta plus cosine squared theta over sine squared theta. Do you know
your identities? Sine squared plus cosine square is equal to one and any time you have
one fraction over one fraction you can flip that bottom up. We are going to flip the bottom
up and write two cosine theta over sine theta times sine squared theta when we flipped the
bottom up, and don’t forget that sine squared plus cosine squared equals one. The sines
will cancel out. This one sine in the denominator will cancel out with one of the two sines
in the numerator. Remember you cancel out factors but not terms. We get two times sine
theta cosine theta. I am just turning the multiplication around to perfectly match the
identity. That is ok because three times five and five times three is fifteen. This is the
sine of two theta. Now, if you are really at proofs and you just stumbled upon this
video or you are just getting used to them, or you are using this to teach yourself how
to do these. There is a reason the back of your book says that proofs may vary. Because,
well they may vary:) That is why like teaching trig proofs so much, because there is no just
use these five steps and if you follow them in a certain order you get the right answer.
You have to think a bit more. Play with the algebra, do some substitution, see what helps,
and for goodness sakes if you have a pythagorean identity that is going to help you right off
the bat…maybe you should use it. So, I have shown you a proof where I immediately went
into sine and cosine and I did an ok amount of work. That is fine and I get the right
answer. You know what, this is full credit…BAM! Well, here is another way. Somebody might
be going, why didn’t you do it like this? And you are going to go, because did it MY
WAY!!! One plus… I know I am bad at singing:) One plus cotangent squared, let’s write this,
two cotangent theta over one plus cotangent squared is cosecant squared. Now when I turn
everything in into terms of sine and cosine, I will not have to find common denominators
and I will be done much quicker. This is still going to be two times cosine over sine, so
it is two cosine theta over sine theta. Cosecant squared is one over, it is one of the reciprocal
identities, that is one over sine squared theta. Look, see I originally had sine squared
plus cosine squared there after finding common denominators and after you applied that first
Pythagorean Identity, so I kind of needed to know those Pythagorean Identities any way
regardless of how I do this proof. Any way, I am rambling. The denominator is going to
get flipped up becoming sine squared theta over one. Once again, the sines will cancel
out and once again we have the sine of two theta. So, I did one proof two ways. Is that
two examples or is that one? I don’t know:) But, when you are doing proofs, when you are
verifying identities you want to… Well, look at the two sides and see what is going
on, are the angle measures the same, are the degrees the same. The only way you can change
the degree is through factoring and canceling (or Power Reducing Identities). The way you
can change the size of the angle measures is the substitution of the half angle or double
angle identities (or sum and difference identities) or at least that is all I am covering. See
if there is any Pythagorean Identities or any of those heavy duty identities whether
it is sum or difference, half angle or double angle, power reducing, whatever that is going
to help you right off the bat, then go to turning everything into sine and cosine…not
as a last resort because that is a really common skill or process. But sometimes you
can shorten your proof up by applying some identities early on if there is any there
to help you. Ok, I have a feeling this is going to be a two part lesson for proofs.
How about we take a look at the cosine squared of theta over two, and let’s see if we can
prove that this equals the secant of theta plus one all over two secant theta. Well,
cosine of theta over two is the square root of one plus cosine theta over two, not anything
to do with secant. So there are no identities that are going to help me immediately, no
heavy duty identities, there is no algebra to do, so we are going to have to go into
sine and cosine and hope that the algebra allows us to get the two sides closer and
help this click so that we can see what is going to happen. So secant is equal to one
over cosine theta. One, I am going to make common denominators now. One is going to be
rewritten as cosine theta over cosine theta, common denominator, over two times one over
cosine theta or just two over cosine theta. I am going to put the top together and get
one plus cosine theta all over cosine theta over two over cosine theta. Any time again
you have one fraction over one fraction, you want to flip that denominator fraction up
next to the numerator with multiplication. Any time your denominators are an exact perfect
match, they are going to cancel out. So, we have…wrong color…one plus the cosine of
theta over two. Hmm. Ok, so you could stop there or you can keep manipulating. If you
are not sure how to make this equal to that and you are totally stuck, then maybe just
playing with the other side. You know, what is the cosine of theta over two? Well, it
is the square root of one plus cosine theta over two, and then there is a power of two
up here. See it? So that means that it is the square root of one plus cosine theta over
two in that square root, squared. Well, that power of two is going to cancel out the radical
and if your teacher wants to only see the work on one side of the proof, then you can
sort of manipulate both sides on your scratch work, then anything on this other left hand
side you can go in reverse on the right hand side until you get cosine squared theta over
two. As far as I am concerned, the left and right hand sides match and I am done. BAM!!!
ALL RIGHTY THEN, let’s do another one:) How about one that is pretty straight forward
and then we will do a hard one. Tangent of x over two equals, you might think that these
are all hard, cosecant of x…if this is your first time learning them…minus the cotangent
of x. Ok, if you look at your formula sheet, you will notice that the tangent of a half
angle identity only has sine and cosine in the three versions of that identity. So, no
cosecants and cotangent and we are going to work on the right hand side. Remember, and
also I have not said this in this video, when you are work on proofs you always want to
simply. Work down the tree, work your way down to the main trunk. Don’t things more
complicated and have a thousand branches you can go up to. Ok, whatever, we are going to
make this in terms of sine and cosine. Cosecant is one over sine x. Cotangent is cosine x
over sine x. Well, shoot…we are done. One minus cosine x all over sine x is one of the
half angle identities for tangent. Also don’t stop until the two sides are an exact match
please! Do work your proofs straight down using nice and clear hand writing, no scribble
all over the place. Your teacher, your test papers are giving you the start and finish
of the problem, we have to grade everything in the middle. We have to see every single
step, make sure that your algebra is right and you logic is correct. We are not just
looking at the answer. We gave you the answer. Make sure that your test is gradable, otherwise
you may not like what you get back. Ok, at least my kids won’t. I am not going to try
to decipher their hieroglyphics. The sine of three x over the sine of x
minus the cosine of 3x over the cosine x and
all of this is supposed to equal two. I hope I don’t have to erase anything on my board
because this is going to be a lot of writing though not a very long proof. Well I guarantee
you that some of my students are going to want to say that 3x over x makes 2x, or the
x’s cancel out, or the sines cancel out. This is not sine times three times x, it is not
cosine times three times x, the cosine and sine are math functions. It is the sine of
(3x) and the cosine of (3x), and no nothing is going to cancel. Alright, so get that out
of your head. We want to prove that this equals two. Man, when I look at my formula sheets
I don’t have three x’s. I have half angle and double angle identities. So, this must
also include some sum and difference identities. I am going to rewrite this so that it is the
sine of x plus two x, that is equal to the sine of three x, minus cosine of x plus two
x, and our denominators of sine x and cosine x. I am not even going to mess with the right
hand side. I think I am going to need a lot more room, so I am going to work down and
possibly need some room over here. I am going to apply the sum identity for sine. It is
the sine of the first times the cosine of the second plus cosine of the first times
the sine of the second minus, don’t forget the parenthesis because I am going to write
all this…I am not going to do that right now…all of that is over the sine of x. This
is minus, applying this sum identity, cosine x times cosine of 2x, don’t forget with your
cosine function with sum and difference identities to change that middle sign, so minus sine
of x times sine of 2x all over cosine of x. Now I have got two humungous fractions that
need to come together and clearly a lot of stuff is going to cancel because the answer
is two, right. We need a common denominator so I am going to multiply this entire fraction
by cosine on the top and bottom
and sine x on the top and bottom. WHOOO!!! Man, now I have to write an even longer sentence.
Ok, so we have…let me get down here… we have cosine x times sine x times cosine 2x
plus cosine times cosine is cosine squared x times sine 2x. Now I am making common denominators
here, so I am going to put a minus sign and a parenthesis because there are two terms
in the second numerators. Sine x times cosine x times cosine 2x. I am getting really quiet
here because I don’t want to make a sign error here. All of that is over cosine x sine x.
Ok, now what do we have here. We have cosine x sine x cosine 2x, cosine x sine x cosine
2x, this one is positive and after we distribute the negative sign that is going to be positive.
So, cosine x sin x cosine 2x is going to cancel with minus sine x cosine x cosine 2x. That
is going to save some writing! We got cosine squared x times sine of 2x. I am going to
need to go up here now. I hate jumping around like this. I hope that you can follow it.
Cosine squared x times
sine 2x, negative times negative is positive sine squared x times sine of 2x all over the
denominator cosine x times sine x. Good thing… I might still need to erase part of this.
You might want to copy what I am doing here. Got it? Ok, so I probably should erase the
holiday lights for this. We have got the sine of 2x and the sine of 2x in both terms and
we are going to factor that out. We are going to pull this out and get the sine of 2x times
cosine squared x plus sine squared x. This is all over cosine x times sine x. Are you
starting to feel it. I am getting excited! We are almost done. Cosine squared x plus
sine squared x, that is the first Pythagorean Identity and that is equal to one. Now we
have the sine of 2x over cosine x times sine x. That becomes…. We have a double angle
and that is a single angle, so nothing is going to happen until we fix that. This is
sine of 2x is two sine x cosine x all over cosine x sine x. The cosines cancel out, the
sines cancel out, and I just did all that work to get an answer of two. Are you kidding
me?! BAM!!! Mr. Tarrou. Go Do Your Homework!

100 Comments

  • Karina San says:

    I am the one who is glad I found your channel, of course I will share your vids to any friends. Thank you ^_^!

  • ProfRobBob says:

    Great job on that 90 grade by the way…And THANK YOU for sharing:)

  • Syed Ijlal Ayub says:

    i'm searching topics about domain and range of sine and cosine…

  • ProfRobBob says:

    I would start with my video about understanding the basic graphs of sine and cosine and/or look at graphing sine and cosine without a calculator. The domain of Sine and Cosine is all real number, so you only have to worry about range. If you know the amplitude and the vertical shift you will be able to figure out the range.

  • Syed Ijlal Ayub says:

    okay thanks.

  • Outofit says:

    BAM!!! When hes done you want need anymore chalk!!!!

  • ProfRobBob says:

    Don't worry…I have chalk for years to come!!!
    Thanks for liking and subscribing too:)

  • ProfRobBob says:

    I hope you went into class and passed that exam like BAM!!!

  • Gregory Paul says:

    This is great stuff! We need more teachers like you. Keep the lessons coming, you've been saving my butt in trig class.

  • ProfRobBob says:

    Thanks for the compliment and THANK YOU for choosing my channel to learn from! I hope you are sharing it with all your friends and classmates…remind them that it is important to like and subscribe to help educational channels like mine to groW and remain FREE!

  • DORC101 says:

    Thank you sooo much for this, I really wish you were my math teacher… My teacher is boring as hell… 🙁

  • Wolf Mobile says:

    Hahhaa,,, thank you! I've got a test today and I this helped a bit!

  • Ronita Finch says:

    Test tomorrow. ..this helped…I think I would've cried trying to do that problem! Thankssss

  • Zayd Ali says:

    You're such a bro. This helped so much

  • LumpinLump says:

    I really love when he brakes his chalks like that at the ending of his vids haha

  • Anoir Trabelsi says:

    We need IMO

  • William Hazlehurst says:

    The last question could be done much easier boss by taking sinxcosx. as denominator, the numerator will be sin3xcosx-cos3xsinx= sin (3x-x)= sin 2x. The bottom sinxcosx=1/2 sin2x,
    Sin2x/ 1/2sin2x= 2 QED

  • RamJim9697 says:

    i keep replaying the smashing of the chalk at the end lolol xD thanks!

  • Nick Briel says:

    You have helped me remember everything I forgot over the semester and now I'm not so worried about my final exam

  • Frankie Lanzana says:

    your videos are awesome… I'm taking an online trig class and strictly watch your videos to study. Throw that chalk and walk off like a boss!

  • Alejandro Cano says:

    AMAZING VIDEO!!!!!!! (sorry for the caps) LOL

  • Alex Mandel says:

    Did all that work for an answer of 2?!?!?! destroys chalk on the board I just about died laughing, but your videos are helpful, but I feel like the practice problems might be too easy.

  • Ivan Anyaegbu says:

    Great videos….they always help me. Thanks

  • Michael Starns says:

    About to rock out on today's trig exam.. like BAM.

    Thanks as always Prof. Rob.

  • Houston Orr says:

    Thanks tons for all of your help!  Feels like I have my own personal tutor at the click of a button!   Really helps me grasp the information.

          Appreciate all the help Prof. Rob. You da best! 

  • Scot Matson says:

    Another chalk throw and a "do your homework" jingle! These videos keep getting better and better. Just tested today and felt extremely confident with the outcome. Really enjoyed the meaty proof you closed out this tut with, I think my teacher goes a little easy on us sometimes. Wish I had a instructor like you in High School. Thanks again PRB!

  • Juan Ochoa says:

    I wish you could be my teacher! I learned more in this 17 minute video than i did 2 weeks in my Trig class.

  • kenzweiler says:

    You should teach professors how to teach/instruct/lecture. Your videos are never boring and never lack enthusiasm or value. Could you do a video on this stuff using sec,csc, and cot

  • Nathaniel Mauga says:

    BAM just aced trig final because of these awesome videos, now on to calc

  • Scott111188 says:

    After converting the top fraction and bottom fraction into cos's over sine's I multiplied the top fraction and the bottom by sin^2x and did the cancelling until I got 2cosxsinx/1. Did I follow the laws of algebra correctly??

  • Gully Squad says:

    Did anyone else catch the Ace Ventura reference? haha 

  • LAWRENCE BAISLEY says:

    I hear you say, "Go do your home work, " after several of your videos. Is there any practice questions somewhere?

  • Metzeler Tumamak says:

    Wish you could be my teacher!
    Our teacher is too fast and doesnt set specifics and is quite rather trivial.

  • Ramon Santiago says:

    @ProfRobBob Didn't you get rid of the negative sign in between both fractions? Why so you distribute the negative sign when you move the equation up?

  • Nawreez Hubail says:

    nehahahaha u r amazing

  • HerpDerpMapleSerp ' says:

    You are AWESSSSSSSSSSSSSSSOMEEEEEEEEEEEE. Please come teach at Citrus College here in California. LOL . 😀

  • Eveling Escober says:

    Great video! Keep it up,I need you jeje 👌👍👏

  • Jake Burnett says:

    OH MY GOSH THIS VIDEO GOT MY HEART RATE GOING ALL FUNKY WOW PROOFS ARE AMAZING I LOVE LIFE

  • Corey Crick says:

    I bet you go through a lot of chalk lol

  • Negin K says:

    You are an actual life saver :)) i was worried sick that i would do horrible on my trig test but after watching your videos for 2 hours i feel like im finally prepared. Thank you so much!

  • L Xiong says:

    You make me cry thank you so much!!

  • Newbport says:

    hahahahaha

  • Sai Ram says:

    how to prove:  tan3xtan2xtanx=tan3x-tan2x-tanx

  • Pranav Mutyala says:

    really helped

  • Fudge cake says:

    wooow woow woow.. IF U'RE MY PROF, I WILL NEVER FEEL SLEEPY THEN! YOU'RE THE BEST! #BAM!! xD

  • Ryoung Kim says:

    Wow! Thank you for showing us every single step! I'm about to ace this test that I'm taking in few days! #BAM hahaha

  • Matthew Cropper says:

    I like your videos but that's not the right use of the word bam or bayyyym if your American. It's not necessary.

  • Eric Murphy says:

    Awesome Professor Rob, as always

  • Brenda Telles says:

    Holy crap! Best teacher out there. Im in college taking trig as a pre req and my professor is the worst. I've used these videos to teach myself and have made A's in all my tests:) thank you!

  • Scieneering says:

    17:27!!!! I wish we had more teachers like this!! You really do put the joy in learning math!!! Thank you again for all your work!

  • Parnika Kapur says:

    For the first example, I saw 1 +cot squared x as cscx. Thanks for this video. I just love trig identities. It was great learning half angles from you as I barely understood it when I first learned it.

  • Judith P says:

    Thank you so much for patiently explaining everything, Sir. 🙂 There are no students who are terrible in math but are simply oblivious of what they are supposed to do since the professor won't directly tell the guidelines in solving such. I'm surviving my semester with your videos!

  • Sameerah Lacson says:

    You've just made my day! I was struggling with the cos^2 (x/2)

    I tried to figure it all out by myself but ended up crying (not really) but BAM! I saw this vid and thanked the math gods for you sir! cheers!

  • s says:

    "I'm not gonna try to decipher their hieroglyphics." Hahaha you're the best xD

  • Logan M says:

    "cosines cancel out, sines cancel out.. and I just did all that work to get an answer of TWO!? Are you kidding me!? smashes chalk WOOO BAM!!" hahaha, i laughed so hard.

  • ProfRobBob says:

    Half angle and double angle Trig proofs is now Closed Captioned!   #math  

  • Searia Kett says:

    I DO NOT MISS high school at all. I can't believe I used to know this enough to graduate with honours. Well kids I went on to University and had a successful career and never used this stuff again. You however are learning it NOW. If you want a career in computers or the sciences YOU WILL NEED THIS. Work hard and remember it.

  • Kristian Factora says:

    i got a different approach on this proof also but BAAAAM! go to your homework haha..

  • Anaflor Cabalda says:

    Hi Prob Rob Bob, I would like to ask this cosx(sinxcos2x+cosxsin2x), why did you not multiply cosx with cos 2x? and also cosx with sin2x?

  • Anaflor Cabalda says:

    Your videos helped me a lot. I've been watching almost all your trig videos and BAM! You're great! Thank you.

  • Erwin Jed Racho says:

    i also smash my pen just as what he did when i solved one of our identites but i got kick out of my room :'(

  • Ronaldo Ramos says:

    how did the 2cossin = 2sin?

  • Trevor Holton says:

    Dude your a lifesaver. Very easy to understand as well as the fact you are entertaining to watch and as a result, i stay engaged in what you are teaching.

  • Jacob Dominguez says:

    Excellent video.

  • James Lloyd says:

    Smashing the chalk was your best exit yet.

  • member762 says:

    drop the mic, put you cape on, and walk off the stage.

  • mimi fu says:

    How does tanx/2 = 1-cosm/sinx? What identity is it?

  • Noor H says:

    Hello there! I wanted to let you know that I watched all of your videos on half and double angles and sum and difference formuals to prepare for a quiz in precalculus. Thanks to your videos, I thought the quiz was very easy, despite the number of students in my class who thought it was our hardest quiz all year. I will let them in on my secret haha. Thank you for your videos 🙂 I am so happy I was able to find the best math content on YouTube.

  • Michael Hanach says:

    How many sticks of chalk do you need to buy to replace the shattered ones? XD jk… But you videos really do help… I have a test tomorrow and i've been watching your videos for the past few days, and I'm feeling confident. Thank you for helping me learn trig! I couldn't have figured it out without you 🙂 Keep up the good work because it really is helpful

  • ralph nagtalon says:

    thanks you for this video 🙂

  • JDNATE says:

    5 years and it still helps me and I've never seen a teacher so excited about math lol

  • tenton2000k says:

    At 12:17, why did you split the 3x into 2x+x and when do we do this?

  • guy next says:

    you make math a lot of fun , you changed my perspective of mathematics since I started watching your videos ,thank you very much

  • Jacob Koman says:

    I wish I had a math teacher as enthusiastic as you are in school. Thanks for making videos! I really like how you take the time to explain what you are doing and why. Subbed!

  • robin grace says:

    You are a math God. Thank you for these! They are saving me currently. Nothing this semester has made as much sense as your videos do. I think I might just watch these in class instead??

  • Minoush Niknam says:

    I love your lecture. I whish you were my teacher

  • Jonel Almario says:

    "I just did all that work just to get an answer of 2, are you kidding me?"
    -Every highschool student ever =)

  • Fro Mra says:

    BAM!!!!! i wish my 75 year old teacher smashed chalk into smitherins against the board with a BAMM!!! She's a good teacher… but she's up there in age if you know what i mean.

  • c M says:

    That ending was boomshot, you have been very helpful keep it up

  • Jack Romer says:

    Thank you so much! I lost my notes and my exam is tomorrow you saved my grade!

  • Kush Patel says:

    My teachers would be jealous that you have enough chalk that you can just throw at the wall 😂😂😂. Anyways amazing job on the videos! They are a live saver for my Pre-Calc class. I don't know what I would do without them!

  • baela U says:

    Epic ending! haha

  • sara abbara says:

    Love your activeness

  • sara abbara says:

    Hahahah you're so funny

  • Livan Ramirez says:

    Is the proof of sin2theta=2sinthetacostheta just sin (theta + theta) or is there more to it?

  • Bashir Gagigi says:

    I would like to take a moment to thank you for your fantastic work sir !
    I depend on your explanations very much and your precious help is very highly appreciated by me !!
    Absolutely phenomenal !
    Keep it up !

  • zelda64rules says:

    Thought I'd make a proof of my own:
    tan (a ± b) = (tana ± tanb) / (1 ∓ tanatanb); tanx = sinx/cosx
    This one was tricky at first.
    Prove: (tana ± tanb) / (1 ∓ tanatanb) = (sinacosb ± cosasinb)/(cosacosb ∓ sinasinb)
    Multiply by secasecb/secasecb
    = ((sinacosb ± cosasinb)(secasecb))/((cosacosb ∓ sinasinb)(secasecb))
    Distribute
     = (sinacosbsecasecb ± cosasinbsecasecb)/(cosacosbsecasecb ∓ sinasinbsecasecb)
    cosxsecx = 1, where cosx =/= 0; which shouldn't be a problem in this case given quotient rule
    = (sinaseca ± sinbsecb)/(1 ∓ sinasecasinbsecb)
    sinxsecx = tanx (which also works inverted: cosxcscx = cotx); What I call the trigonometric product rule
     = (tana ± tanb)/(1 ∓ tanatanb)

  • RITIK GUPTA says:

    You're Great SIR . You made me understand the graphs of trig functions . i was not able to imagine about the domain and ranges but you are the one who cleared my concepts . You take your precious time out to share your knowledge with us which deserve a applause. We all RESPECT you. I Subscribed your channel and wish to see more new videos from you Sir!

  • BucsFan10 says:

    Sin^2(theta) +sin^2(theta) = sin^2(theta) ? This is at 1:52 and I am confused how you added sin^2(theta)/sin^2(theta) and got what you got.

  • Hemant Narasimhulu says:

    As a High School Student sitting for Cambridge International Exams, Sir you deserve more than a like button after 3 long years I finally understand this part of trigonometric Identities and now i will follow this path in exams and continue to work hard….Your explanation are simple and very promising…Keep it up, you will make your way even higher than expected SIR…..

  • Benjamin Lehman says:

    I have a problem with 7x can I split it to 3x+4x or do I have to split is up more?

  • Sergio Campos says:

    That chalk bit is the equivalent of smashing a guitar after rocking out

  • Alexander Colgan says:

    Love the chalk explosion at the end 😂 now I won’t fail the test tomorrow!

  • Adin Boerman says:

    Thank you for the amazing video!! I am so grateful that you took time to show me how to do analytic trig! I have a test this monday and this helps me feel prepared.

  • Catherine Hernandez says:

    man watching this is like watching magic I swear

  • Gavin Ganzorig says:

    You make it look too easy.

  • Programmer Rules says:

    Bam😂😂😂

  • Priscah Tetio says:

    Wow this is so Amazing,,,

  • The BY says:

    Bro..for the last question i just found a simpler way.
    Sin 3x= -Sin x and Cos 3x= -Cos x
    So i subtitute it into the equation,
    -Sin x/Sin x – (-Cos x/Cos x)
    -1-(-1)
    2

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