Trigonometric Equations Multiple Angles 0 to 2pi Restriction
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- Опубліковано 7 вер 2024
- I continue my examples of solving algebraic equations that involve multiple angles.
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I know it's been over two years since this video was made, but I just wanted to say thanks for finally helping me understand multiple angles. I'll definitely be checking back to this channel whenever I need help in mathematics.
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My teacher never mentioned how the coefficient is how many times you go around the unit circle (if he did, I wasn't paying attention), and that really helps. Now I understand this way more! Thank you!
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If it is of any help: To answer the question (using Mr Tarrou's first example) of why we get what "appear" to be negative "answers" for the evaluation of cos-1(1/2). In the UNIT CIRCLE our answers for the function cos-1(1/2) would be pi/3 and 5pi/3, - ONLY - that is correct thinking.
HOWEVER, what we are trying to do here is find all angle values that get "cut back" (divided by the coefficient in front of x) to fall within the unit circle (that is, between 0 and 2pi).
In the first example: The function cos-1(1/2) yielded: pi/3 and 5pi/3 (both positive x-axis values). These then get cut back (by 1/2) to give pi/6 and 5pi/6 (a negative x-axis value).
Because we are cutting back, this means that there "could be" some other angles out there (greater than 2pi), that when cut back by (1/2) also fall in the unit circle. Mr Tarrou then produced 2 more angle values: 7pi/3 and 11pi/3 (both positive x-axis values). These then get cut back to ( by 1/2) to yield: 7pi/6 (a negative x-axis value) and 11pi/6.
So the answer contains two angles 5pi/6 and 7pi/6 that "appear" wrong at first sight, but really are not.
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It is if you are working within the restrictions. But if the question asks for all answers between 0 and 2pi or just all answers you need to go beyond just the restricted values.
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The 7pi/3 came from the fact that I must go around the unit circle twice because of the double angle. I'm continuing to count by pi/3 until I get back into quadrant I again (7pi/3) and IV again (11pi/3). Same is true for the 8pi/3 for the next question but I needed to get back around into quad II in that question.
Professor RobBob, thank you for another well planned video/lecture on Solving Trigonometric Equations with Multiple Angles that has Zero to 2 PI Restrictions. Students should use the examples as guide when doing problems.
Thank you. I have been stuck for two days with no help. You have saved my life.
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I really appreciate all the videos. I like your technique on explaining multiple angles in a trig equation. My question is on Question #3. If you graph the function, x=0. This value was not found algebraically. If you check the function with x=0, it works. Just not sure how to account for the zero in the computation besides finding it in the graph.
Mr. True...I have been stuck dealing with this type of equations. Thanks a lot!
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Keep watching and learning!
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Single angles like tan(x)=.... or multiple angles like sin(3x)=.... or both. My guess is the first. The video will show up Thursday or Friday by the way:)
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Thank you so much for helping and saving my life. You are much better than my teacher. I will make sure to tell everyone about your channel. Btw, do you have a video about sketching trigonomentric functions?
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Yes...just go to my website at www.profrobbob.com/trigonometry/graphs-of-trigonometric-functions
I LOVE YOU MAN! I PASSED MY FINAL EXAM! TY!
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I worked through your question and got the same answers as well as 2pi which is outside the restricted domain...but 0 is and it is in the same position on the unit circle so if you check 0 it works. If there is another technique other than this...I have forgot it:( I hope this helped. The trig in my PreCalc book doesn't offer any questions with this kind of issue that I can remember.
Made me understand this concept far better than my precalc teacher. Thank you!!
You're welcome.
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Thanks for using the phrase" Cos of 2x"...........just hearing you say "of " made a huge impact in trying to solve trig equations.....JJ
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You are awesome man! I was struggling with these so much ty.
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Thank you so much for reviewing this for me so I wouldn't fall behind in my calculus class!
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SIR I AM LOVING TRIGONOMETRY BECAUSE OF YOU. BAM!
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Hey thats my older sister ermily
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+Sam Cantaloup thanks!
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This video was so helpful and fun!
Glad it helped!
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I don't believe I have a video with that example. Is that cos(2x) or (cos(x))^2? You will want to start by writing the equation all in terms of one trig function and with the same angle measure. If a double angle, start with a double angle identity for cosine that only has the sine function on the right side of the identity.
Before I start making notes on this...Do you mean questions that ask for all possible answers and not just between 0 and 2pi?
I know this video is older but I'm currently learning this and this was super helpful! The only thing different is my teacher wants us to use the format of answers + 2 pi k, where k is an integer. It's a lot more confusing for me to figure out how to write those intervals than to just go around the unit circle and list all the rest of the possible answers.
Jasmine Gault It sounds like you are looking for all solutions, not just those that are between zero and 2pi:) Thank you for the kind words and sorry for the slow reply.
THANKS!
Thanks, you made this simple. The text book made it seem more complicated than it really is.
You're welcome Nelson Villafane and thanks for subbing, I appreciate the support!
you have beaten khan academy, your way of explaining is awesome love the way how you keep balance between entertainment and study you are good at getting our focus and BAM!!!!!!!!! gotta do my homework :)that's the end. keep smiling
+Najeeb Tareen BAM!!! ...now if we could just get the same amount of subscribers that they have:)
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Thank you sir. I think I am getting somewhere. Clear explanation indeed.
No, that was just an middle step of me trying to show all the work required to get the final answers of x.
Thank you for taking the time to explain things in a way that actually makes sense! Keep the videos coming... Could you please also make some General Chemistry videos? lol
Thanks for watching and subscribing but you will only get math videos on my channel:(...sorry, I'd have to go back to school to do that...lol
Are sin2x and cos 2x both double angles or are one of those 2's supposed to be an exponent like (sinx)^2?
extremely helpful video. keep it up!
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Are you trying to find an exact answer to the sin(10 degrees) without a calculator? If that is the case I don't know how to help you. I only work with half and double angle identities and sum and difference identities in my class. Neither one of those will help you find a solution in exact form.
You're welcome!
Really wish you were my trig teacher. I'm doing so horrible right now and I'm almost done with the class. Here's hoping these videos for the next lessons will help and help me pass with a C >.
Go ahead and just aim for that "A" and study till you can't study any more and then watch it one more time, the next thing you know, you'll be jumping up and down because you "got it"
cannot thank you enough!!!!!!!
Sub and share...that's thanks enough...BAM!!!
THANKS:) It's good to see your face again!!!
You;re a good man Mr T................Thank you-----JJ
THANKS again john james
...and you're welcome :)
your a awesome instructor!
You're such a James Spader look-a-like in this video, I love it haha :) keep up the great work Mr. Tarrou
Must be the longer hair…as long as the reference is one when he has hair or does this mean I've been "blacklist"ed…lol
Thanks man!
You are very welcome. Thanks for choosing my math videos to learn from😀
Realllly thank you you have cleared thousands of questions
But there is a question that I asked for no one answered me pleaaaase explain it for me,you know one question would make the whole chapter not understood,
The question is and stay with me step by step ,didn’t we say that there is restriction for the inverse of cos and son and tan ,for example the restriction of the inverse of cos is 0to 180.Right...
So why we didn’t restrict that here to it is an inverse so why not
Pleeeeaaaaase answer me
And I am so thankful to you
You're welcome and thanks for watching this channel!
I hope you got your question answer by now...I'm Mr T's wife and only answer general comments since Mr T has such a heavy work load at school this year. We do appreciate your support and hope you will sub and share this free resource with everyone...BAM!!!
Thank you so much!!
+Tinpoppy you are so welcome, thanks for watching...please like, sub and share:D
Awesome as always!
Oh I see you got the same angles at sin(2x) because in your example cosine is multiplied by two. ok okn :)! hey btw great videos! my book makes everything more complicated and you have really make it way much easier... do you have any videos on solving composite functions? i have problems solving them.... i know the process but i get stuck in some steps... thanks!!
Thanks so much!! I do have one question though ... on example #3 (cos2x-pi/4)=root2/2 ... which are the actual answers? There are three lists of answers, would I include the middle list where you added pi/4, or is that just an in between step to get to x?
COOL TEACHER! THANKS FOR AN AWESOME EXPLANATION!!
+Luis torres thanks for studying and subbing!
Great video as always. I have been learning so much from your videos. I have a question @ProfRobBob. What if I had sinx 2x + sin 5 x. How would I approach this problem? And ifi HAD SINX 2X + sin 6X . How would you set it up? Would it be 2sin 2X.cos2x. Thank you
late answer, but you can use the fact that
sin a = sin b =>
a = b + k2pi
OR
a = pi - b + k2pi
for all integers k.
similar arguments can be made when you're dealing with cos or tan or whatever. you can derive them easily by looking at the unit circle and thinking about where the functions are in it.
if youre dealing with something like cos 3x = sin x, then make sure to first "convert" sin to cos or the other way around, using some known identity for that.
thank you soooooooooooo muchhhhhhhhhhh... THis was very helpful!!!!!!!!!!!!!!!
can you help me with this one problem? find all angles on the interval [0,2pi)
cos(2x-(pi/6))= sqrt3/2
I did all the step like how you solve in the videos, I came up with pi/6, pi, 7pi/6
But the actual answers are: 0, pi/6, pi, 7pi/6. Can you tell me why?
If they are all double angles, can you send me a personal message through my Facebook fan page. I am going to check my answer and if it is correct, it will be much too complicated to explain in this comment box.
Hiya :) I don't quite understand what you say when you mean go all the way around the circle. For example, in the example problem one, I see where you pulled the (eventual) pi/6 and 5pi/6 from, but where did the 7pi/6 and 11pi/6 come from? Any insight would be appreciated. Love love LOVE your videos by the way. You're a life saver!
Remember that 2x means to go around the unit circle twice. First he got pi/3 and 5pi/3 from cosx=1/2. But since it is asking for 2x, you have yo add 2pi (6pi/3) to both. pi/3 + 2pi = 7pi/3 and 5pi/3 + 2pi = 11pi/3. The last step is to multiply everything or both sides of the equation by 1/2 to isolate 2 from 2x. Hence, the answers are pi/6, 5pi/6, 7pi/6, and 11pi/6.
to isolate *x from 2x
thank you soooooooooooo muchhhhhhhhhhh... THis was very helpful!!!!!!!!!!!!!!!
ProfessorBob, what is the reason the restriction is written as [0, 2pi) instead of [0, 2pi]? a lot of examples I have seen in the textbooks use [0, 2pi) instead of [0, 2pi) without an explnation and proceed to solve the problem. It seems that there is no difference between [0, 2pi) and [0, 2pi]. In your example, you ended up with one less answer.
0 and 2pi are coterminal angles, thus 2pi is the first angle of the second revolution around the unit circle.
thank you professor Bob.@@profrobbob
@@golagha1959 My pleasure.