Rational Trigonometry Puns

I've been looking at the equation

[; \cos{\left(\frac{i}{n+1}\pi\right)} + \cos{\left(\frac{j}{n+1}\pi\right)} = k/2 ;]

where i and j are integers between 1 and n (inclusive), and k and n are integers.

Specifically, I've been stuck on showing that for k=2, the only solutions are when i=j=(n+1)/3. I can show using Niven's theorem that this is the only solution when both cosine terms evaluate to rational values. However, I'm stuck for what to do if one cosine term evaluates to 1/2+m and the other to 1/2-m for some irrational m.

The only possibly useful information I could find, was this paper that describes how to find the degree of algebraic integer for any rational angle. I think that if the cosine terms are both irrational and sum to k, then their degree must be 2, as the only identity I found that fits this description is for k=1: cos(pi/5)+cos(3pi/5)=1/2. However, I have no way to prove this.

Any thoughts on what to do?

Made a post earlier about how I wanted to start learning mathematical concepts, and someone suggested looking into Rational Trigonometry. Looking online, lots of criticisms and praise, not sure what to think (since I'm too ignorant on the subject to draw my own conclusions).

Is Rational Trigonometry legitimate, or should I not waste my time watching the videos and reading?

So I couldn't understand my instructor very well, she was going through things so fast, I want to work as hard as I can.

So we were doing a review, and when it came to adding and subtracting rational expressions I was completely lost because my instructor kept talking and moved on so quickly.

So anyways, I really need help with this.

So I'm watching Khanacademy's video on this topic, this is his equation:

http://prntscr.com/89pnmu

During class they were saying how we have to find the LCD, this is where shit got me confused. The instructor always picked the largest number. I got so confused I lost track of what the fuck she was doing.

So Khanacademy, he says at around 1:25, the LCD needs a 2, X+5, X-5, and X+5 * X-5.. Why do we need all of those numbers? How are they the Least Common Denominator? Please help.

Okay I’m rewatching from the start for like the fourth time (I haven’t been this bonkers about a show since twin peaks came back).

Re: Tai and Shauna’s convo in ep 7 that Nat is the reason they’re all alive and the general assumption that the spike pit in the opening of the pilot was created to trap girls for cannibal reasons.

Hear me out:

As winter approaches, Nat realizes that she and Travis can’t be out hunting for long hours and little reward, so she is the one who suggests they create the spike-pit traps in the woods around the cabin.

During the days/weeks of early winter, the girls slowly create their “winter looks” with animals they catch in the traps.

But then the dead of winter sets in, and they continue to check the traps every day, and for a long time every day they’re empty. But they have their furs, they are somewhat warm.

So why is the victim in the opening of the series barefoot and in her nightgown?

There’s a crazy phenomenon that can happen with severe hypothermia called paradoxical undressing. link

Basically, your nerves get so damaged by the cold that you begin to feel hot. like, so hot you can’t handle having clothes on. at this point of hypothermia delirium tends to set in as well.

One of the famous examples of this included in the link above is Patrick Dolan, who in 1847 was stuck in the Sierra Nevada during winter with the Donner Party, a group of westward settlers made infamous after getting stranded on their journey without enough rations to last the winter and (wait for it) resorting to cannibalism to survive.

So my theory is that like everything else that’s been revealed so far in this series, the truth of what happened in the woods is more sad than sensational.

During a bout of hunger- and cold-related delirium, one of the girls (Lottie?) succumbs to paradoxical undressing and escapes from the cabin to get “relief” in the cold. she’s out of her mind, obvs, so she doesn’t consider where the traps are located. (edit to add: in this theory, she hears the menacing “voices” while she’s running bc of the hypothermia-related delirium and, if it’s lottie, bc she’s lottie. her foot is bloody bc she’s been running around barefoot in the goshdang woods for god knows how long by the time we catch up to her).

The girls find her during their morning trap-check, and now they’re faced with a decision: starve to death or resort to the unthinkab

This was given to me in a test review sheet, and it is the only one I can not solve. I am hoping someone can help me out! I know to rationalize the denominator, you time the fraction by denominator/denominator, but it still is not turning out right for me.

Find the exact value of the following without using a calculator:

sin[sin^(-1)(1/6) + tan^(-1)(-5)]

Simplify your answer. Give an exact answer, using radicals as needed. Rationalize all denominators.

I've gotten to this point:

(1/6) + [-5/(√1+5^(2))]

(1/6) + (-5/√26)

(√26 + -30)/(6√26)

I'm teaching my college's introductory physics course for the fourth time. Long story short, it is a fucking mess. Half the students can't do basic algebra or trigonometry. I'm a week and a half behind where I normally am at this point in the semester.

I've slowed the pace of the course and done some review of the math required, but apparently that's not enough. A student has decided to appoint himself spokesman for the downtrodden students against the wicked professor and sent a very condescending email, and I'm honestly struggling with how to respond to it with something other than "Fuck you, drop the course if you don't like it."

According to him, he's talked to "several peers" and they don't understand the material. He finds the fact that I don't slow down to the pace of the slowest students "disrespectful". He says that I am not doing my job, which is to "make sure the students understand the course and the material presented." Apparently I'm "lacking in this responsibility." He wants to "meet and discuss options."

I've never had an email like this before, and I'm a little taken aback by the sheer arrogance of it. Who the hell does this student think he is, to talk down to me like I'm his fucking employee?

If anyone has some insight into how to respond to this email, I'd appreciate it, because I'm sitting on an extremely hostile draft reply and the rational part of my brain is telling me that it's probably not a good idea to send it.

https://preview.redd.it/r10gq9gifi681.png?width=434&format=png&auto=webp&s=11a3f39529f4e53250f1fea9ce4699cbbef5b3aa

I've been deep in a symbol rabbit hole, trying to figure out what the heck it means. I think it's another one of those "there could be a rational explanation or it could be magic." You can find all these symbols - circle, triangle, lines - in both trigonometry and spiritual/magic symbolism. And it just depends on your perspective.

Aside from the rational - in mysticism, a circle contains the magic used in ritualism; a triangle symbolizes fire. So, when the girls created a circle of fire, they unleashed...something. (Lottie.)

One thing I think is really interesting is that sickle/scythe thing at the bottom. I know some people think it looks like Saturn or ceres, and I agree. I also think it looks like the symbol for Lilith.

Here are some things I've learned:

1. Lilith is the original wife of Adam

2. Lilith was an independent woman, demonized for her independence and refusal to be subservient to Adam. Lilith means “the night,” and she embodies the emotional and spiritual aspects of darkness: terror, sensuality, and unbridled freedom

3. Lilith gives birth to powerful children and can revitalize the fallen by rebirthing them. She is known as the mother to demons

I think there is a theme about fertility/reproduction here. I think Shauna's pregnancy will play a significant role in whatever rituals they adopt, and is definitely swaying me to think Shauna is AQ.

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

<-------- Previous Chapter

____________________________

You know what's the main problem with living on a planet that's litterally blown up? Apart from the eternal ice age? It's the storms. But, even worse, were the electromagnetic storms. Those were capable of potentially frying a bot if they hit right.

Luckily that evening it was only a normal snow storm.

A human would have probably died of hypothermia in a matter of minutes out there (would have been a good death at least, snow had become a rarity on the majority of the planets that had been colonized). But that wasn't a problem for the bots. And it sure as hell wasn't a problem for a murder drone. Or, in our case, nine of them.

Three pods were smoldering in the ground, three "medium" concentric craters around them that, in three points, touched each other forming a pattern that would remind any student in the galaxy of very bad times in school studying geometry or, even worse, trigonometry.

The pods were damaged, obviously. The Company would never give its creations a chance to get back to their home-planet. Why? Because there was a small chance that the public could find out. And that would have been oh so inconvenient and annoying. Shutting mouths was costly.

Anyways: the retrothrusters had done near to nothing to slow down the descent of the pods, resulting in the actual thrusters being damaged or simply no longer being there in one case, since the attrition with the atmosphere had taken them away. They were now lying around somewhere on the planet but, good news, they probably hadn't hit anybody!

Inside the pods nine yellow "X"s turned on, fans whirring to life in previously lifeless (well, if you can say that a robot is alive. Oh god, please Isaac Asimov fans don't kill me). Writing started to appear on the monitors that represented the drones' faces, peronality programs were charged and activated, systems and weapons were checked. All was perfectly functional. Finally, eyes appeared on all the monitors and looked around the inside of the pods. They were, for lack of better term, barren. And that is a compliment. The drones could consider themselves lucky that there were actual seats in there and that they didn't have to sit on the metal floor. Not that they would have minded, after all, the Company first created them and then gave them a purpose, why would they have to compla

Hello good people of Reddit, I am a verified MATH expert who has competed over 50 tasks in the past two weeks. Also, i have served as an online math class assistant for two years:I offer my students many services, including but not limited to:Taking up your MATH CLASS; weekly assignments, quizzes, and exams involving:

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Do your worst!

I'm surprised it hasn't decade.

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

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A common refrain I often hear from students who are new to Calculus when they seek out a tutor is that they have some homework problems that they do not know how to solve because their teacher/instructor/professor did not show them how to do it. Often times, I also see these students being overly dependent on memorizing solutions to examples they see in class in hopes that this is all they need to do to is repeat these solutions on their homework and exams. My best guess is that this is how they made it through high school algebra.

I also sense this sort of culture shock in students who:

• are always locked in an endless cycle of “How should I start?” and “What should I do next?” questions,
• seem generally concerned about what they are supposed to do as if there is only one correct way to solve a problem,
• complain that the exam was nothing like the homework, even though the exam covered the same concepts.

Anybody who has seen my comments on /r/calculus over the last year or two may already know my thoughts on the topic, but they do bear repeating again once more in a pinned post. I post my thoughts again, in hopes they reach new Calculus students who come here for help on their homework, mainly due to the situation I am posting about.

Having a second job where I also tutor high school students in algebra, I often find that some algebra classes are set up so that students only need to memorize, memorize, memorize what the teacher does.

Then they get to Calculus, often in a college setting, and are smacked in the face with the reality that memorization alone is not going to get them through Calculus. This is because it is a common expectation among Calculus instructors and professors that students apply problem-solving skills.

But wait! How are we supposed to solve problems if we aren’t shown how to solve them?

That’s the entire point of solving problems. That you are supposed to figure it out for yourself. There are two kinds of math questions that appear on homework and exams: Exercises and problems.

“There’s a difference?” you might ask. Yes, there is. An exercise is a question where the solution process is already known to the person answering the question. Your instructor shows you how to evaluate a limit of a rational function by factoring and cancelling factors. Then you are asked to do the same thing on the homework, probably several times, and then once again on your first midterm. This is a situati

They’re on standbi

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Pilot on me!!

Nothing, he was gladiator.

I've been frequenting this subreddit for the last few weeks as a now Grade 12 IB Student. You may have seen me commenting on various threads in this sub. Apart from the people asking about their timetable, the most common thing I see on this subreddit is people struggling with the math series courses (MCV/MHF/MDM). Today alone I counted 6 posts of this nature. Just from reading these posts, I can really feel the frustration and the despair that comes with trying really hard at something but not being able to succeed with it. I know how much that feeling sucks, and how much it can bring you down. I’ve been there before too when I did SL Math last year. So to all fellow students who are out there struggling this year… I just want you to know that you have this. You can 100% succeed. I have faith in you guys to overcome your challenges and get that dream program, raise your results to get that score you need, bring your grades up from that one bad test you had or whatever your goal is. No matter what parents, teachers, anyone else says, never quit. Just keep swimming, and hang in there. I know some of you guys may have home troubles, mental health troubles, or various other factors keeping you from being at your 100% best. Never for a second think you are alone in this, there are people here standing with you, I am standing with you, and I wanted to make this post to help you out. If my posting this makes someone else’s day better, then for me it's a huge success.

My Background:

I took IB SL Math (AA) as mentioned previously over the course of 2 quadmesters in grade 11. For those who don’t know what that is, its basically all of the grade 12 MHF, MCV (without vectors), MDM math courses, and the grade 11 courses combined into 1 course, so you get everything in one. If you feel like you want to verify that for yourself, the syllabus is here.

When I read these posts about how some people are struggling, I can make a personal connection to that feeling because I was also in the same position at the start of the course. Coming into grade 11 the shift from very light work in grade 10 to the rigour of SL Math really caught me off guard. In the midterm of the first quadmester of the course, I ended up with a 66 which was my lowest mark to date. It was like a slap in the face telling me that I really needed to drastically change something

Hello good people of Reddit, I am a verified MATH expert who has competed over 50 tasks in the past two weeks. Also, i have served as an online math class assistant for two years:I offer my students many services, including but not limited to:Taking up your MATH CLASS; weekly assignments, quizzes, and exams involving: ‌✅Precalculus ‌ -Polynomials ‌ -Rational functions ‌ -Trigonometry ‌ - Absolute value functions and inequalities. ‌ - Any topic in precalculus ‌✅Calculus ( I, II, III, IV) ‌ -Differential ‌ -Integral ‌✅Application in Calculus such in Business ,volume , surface, length. ‌✅ Ordinary differential equations ‌✅Matrices manipulation *Iam familiar with platforms including Blackboard/MyMathlab/MyStatLab/MyOpenMath/MyMaths -Affordable and negotiable weekly and monthly plans catered to the students budget. When you hire me, you are HIRING THE BEST. Contact me on discord: smarttutor#2355 Whatsapp. +1(941)9338675 Email- lenniekuether@gmail.com