5 Practice Angle Relationships: Fill & Download for Free

Great question. The answer is no. The only rational values of the cosine of a rational multiple of [math]\pi[/math] are [math]0, \pm \frac [/math][math]1[/math][math] 2[/math] and [math]\pm [/math][math]1[/math][math].[/math] I think it’s illustrative of the poverty of trigonometry as practiced.Sine and cosine of multiples of [math]90^\circ[/math] are zero and [math]\pm [/math][math]1[/math][math][/math], so they’re rational. Nontrivially, only [math]\sin 30^\circ, \tan 45^\circ[/math] and [math]\cos 60^\circ[/math] are rational. We can pick up a few more angles in other quadrants but we’re really only talking about two triangles: 45,45,90 and 30,60,90.The transcendental relationship between angles and their trig functions makes everything difficult. Very little is rational. The simplest things require infinite processes.In Rational Trigonometry we replace lengths with quadrances (squared lengths) and angles with spreads (essentially sine squared). Much of the infinite and all of the transcendental nature of trigonometry melts away. The usual angles of trigonometry, [math]0, 30^\circ, 45^\circ, 60^\circ[/math] and [math]90^\circ[/math] are represented by their respective spreads: [math]0, \frac [/math][math]1[/math][math] 4, \frac [/math][math]1[/math][math] 2, \frac 3 4[/math] and [math]1[/math][math].[/math] Any angles in triangles whose vertices are lattice points have similarly rational spreads.

The main reason why relationships fail is because one person in the relationship is bad at logical reasoning and structured thinking. Though, most of the time it’s both partners that are really bad at it.For example, one partner gets pissed off, because the other partner has to cancel a date because there was an emergency with a friend. The first partner says “YOUR FRIEND IS MORE IMPORTANT TO YOU THAN ME”.The second partner says “My friend would suffer a lot if I didn’t help him out, 100x more than you missing out on the date. However, most importantly, me spending 30h with you instead of 33h this month, is still 10x more time those 3h with my friend, so you are 10x more important than him.”As you can see, the choice is very simple, it is the moral duty to cancel the date and help the friend. An intelligent partner would never throw this accusation at you. However, most people are so immature that they don’t understand that. This is not just some people, that’s probably 90% of people.How do those 90% of people have relationships then you may ask? There are 3 ways:1/3 of those 90% of people fight a lot and end their relationship over a couple of years.Another third just suck it up and raise children for 10 years until they divorce.Another third category that is also not inclined to logical comprehension simply scale back their needs a lot, and is very understanding that there are differencesWhat about the remaining 10% that are actually good at logical comprehension?They understand complex situations easily, can resolve them with simple logic, and thus rarely fight and never have to make compromises.You ask how do I learn logical comprehension?It’s not that hard actually. Here are some rulesAlways weigh the pros and cons of a situation carefully, and completely leave out emotions whether negative or positive when building an argument. Look at it like at a school essay that you get a grade on based on how well you looked at all the facts from various angles.Always prioritize the needs of stakeholders from most important to least important, and see if this is something that is very important to that person and if your point of view is very unimportant in comparison. If it is, you have no right to fight over this point, since the priorities are skewedOnly believe things with insurmountable evidence. Be extremely critical toward all things with zero evidence, weak evidence and also mediocre evidence. Even be critical towards things with strong evidence and very strong evidence. Stop guessing, believing etc. This is the source of all suffering.Do not have strong opinions. Many times you have probably heard “Have strong opinions, it makes you look “confident”. That’s bullshit. An intelligent person only has strong opinions if there is insurmountable evidence. They have studied the subject for weeks and months and have practiced critical thinking for months and years. For things where there is no clear overwhelming evidence, they can still have an opinion, but they are cautious with their opinion and are still looking to collect more evidence. But if they have collected lots of evidence and have run it by at least five different people, they have strong opinions. Only people with low education and little ability of critical thinking have many strong opinions.Be aware of the major logical fallacies and cognitive biases 90% of people commit hundreds of times every single day. This is not a joke. It really happens that much. If you are aware of these fallacies and know them by heart, your life will completely change. If you know these 3 infographics, you already have an overview of most fallacies that people commit constantly and it gives you a very good starting point.The different levels of the quality of an argument:If you have practiced the content of these 3 infographics and the 5 points I listed above for a few months, you are already smarter than 90% of people, because 90% of people are completely clueless about these things.There are more logical fallacies, but those are the most common logical fallacies that people commit.Only lawyers, psychologists, and philosophers likely know about these logical concepts, because it’s literally their job to only see the facts and block out the noise and maybe some highly trained scientists. However, even most academics are completely unaware of these concepts, even many university professors. These concepts are so powerful, so if you have internalized these concepts, your ability to understand complex problems increases massively.This whole area can be summed up under critical thinking, something most people just can’t do because it isn’t taught at school and also not in many university degrees.However, those concepts are so incredibly important in the age of fake-news, propaganda, anti-intellectualism, where people love to believe things with zero evidence over things with mountains of evidence.This brings me to our last concept today. Compare the strongest piece of evidence for something that you want to agree with.You will see that the strongest piece of evidence for the anti-vaccine movement, homeopathy, climate change denial, flat-earth movement is very questionable and doesn’t hold up to 5 minutes of research. If that is the case, you know that it’s simply false.This is the strategy that pseudo-science uses, they make up 20 different arguments that sound somewhat plausible but actually aren’t if you research them and pair them with lots of emotional speech. “AUTISM”.Now, if you sort all the 20 arguments by their strength, you can simply look at the strongest argument and see if it holds up to 5 minutes of internet research. If you want to be very certain, you can research the 5 strongest arguments that you can find and see within 30 mins if an entire movement is true or completely made up:That’s how all pseudo-science arguments are designed. They overwhelm people with an onslaught of arguments, but if you look at them individually, not a single argument holds true.Only if you look at them as a collective they look convincing because 90% humans are very gullible, but if you look at all 20 arguments individually, you will see that they simply aren’t true.Why are 90% humans so gullible? Because they can’t really focus on facts and block out the noise until they know the above mentioned concepts which usually only lawyers, psychologists, and philosophers are aware of.Luckily, now you are, too.Addendum: Some commenters mentioned that living relationships on rationality is bad because you also need emotions - this is a bias since rationality and emotionality aren’t mutually exclusive at all.Rationality has zero impact on feeling emotions. It just explains why one has these emotions and then you can find the cause for bad emotions and fix them.It’s only a problem when you think you are super smart and think that emotions have no value for whatever reason.That is not smart, will make any relationship fail, and has nothing to do with what was said above.Have emotions, but understand why you have them. That’s also called emotional intelligence. Developing high emotional intelligence will give you wonderful relationships.Have empathy, be sweet and caring. Being rational doesn’t affect any of those. But rationality is the first line of defense against every decision you make or action you take, before letting emotions cloud your judgement.Also, don’t entertain negative emotions, there is just no point. Save that energy for the positive ones and feel them all the way. That’s stoicism.

Usually I would ignore such questions. But after reading answers here, before you make a decision or advice someone else, I thought I'd give you some practical insight.I have seen enough of these cases!Since we are talking about physical relationship, I am not considering scenario where you had non physical relationship.The answer here depends on many factors. Firstly, are you talking of arranged marriages. Then, there are lot of things at stake here. Firstly, you have very less time to interact with the guy before a marriage is settled. Hence you obviously can not bring this point. Even if you bring this up at first or second discussion, it will be blown out of proportion and you will be in embarrassed position in front of your own family members.ARRANGED MARRIAGESo, before even thinking of letting your prospective husband know about it, you need to let your family members know about it. I know. It is very difficult. But again, it depends on family. Will your family accept such news? If not, do not even consider it.Now, if you do tell your husband before marriage that you had physical relation (I am assume sex) then 50% of chance that he will agree if he thinks you are beautiful and talented etc.., . But in those 50%, more than half of them live in idealistic world. They think they can handle it but in reality they can not.I know a scenario where the guy promised over and over that he does not care and after the marriage, he kept bringing up her past. They eventually got divorced.What the guy did was- "oh, she might have already had this experience. May be I am not satisfying her as her ex used to... Is she really over her ex? Or Is she still maintaining contact? Let me skim through her mails. Let me get her contact list... and correlated her story with what really happened and see what things she left out"This is not just the story of that guy but many guys. Some guys may not be so blunt but they do have that thought at the back of their head.So, the question is: DO YOU want your husband to have all those thoughts about you.Guys are mostly hypocrites. I am not sure of girls but being a guy and having seen many guys I can say this with confidence. He might have had physical relationship before but IN ARRANGED marriage, if you DID tell your past physical relationship, chances are that it might back fire.So here it goes for arranged marriage - Here is what most likely to happen if you do revealGuy is chaste and nice - It will break his heart. But he will accept. He will suffer within though.2. Guy is chaste and not nice - He will keep referring to your past, paining you3. Guy is chaste and a jerk - he will make you look like slut in front of your parents and you can not lift your head in front of anyone you know. There is no telling to what extent he goes4. Guy is not chaste and nice - probably he will be okay and might not think about it.5. Guy is not chaste not nice - worse than 2.6. Guy is not chaste and a jerk - worse than 3.My sincere advice is, if possible do not reveal this. It is not cheating if you are really over it and want to start a new life. We all have stuff that we are not proud of. If you think, your relatives, friends etc.., can support you in this, better chose not to tell him. If you think your relatives or friends can harm you or blackmail you with this leverage, give only some details but do not go as far as telling him that you had sex. At least do not do it until 3-4 years of marriage. Because in arranged marriages, love gradually develops. If you tell these things during that stage, it will never bud. After spending an year or two, your husband will know completely about you and will take you forever no matter what (hopefully). If you get that confidence, then reveal it to him. Make sure you keep your husband away from evil relatives and 'friends' till then. They can play excellent role in spoiling marriage.LOVE MARRIAGEIn love marriage, you have chance to explain this to the guy before even accepting the proposal. So there "can" be safer chances. Because you have ample amount of time to observe his response.Notice, if the guy's attitude towards you changed after you told him this. If he is really cool about this, you can be sure. Just do not believe when he says 'I am cool'. Observe and make sure for yourself that he is really loving you for what you are. Do not repeat the mistake you did with first guy. Wait till the marriage.Again, if you really really love this guy and can not live with out him and if you fear that telling him will break his heart and make him go away - STILL tell him. If he loved you, really loved you, he will surely ignore it. But if he did not ignore it, do not try to convince him or something. You can not be happy with him.So my take isIn arranged marriage, wait before you reveal or never even reveal itIn love , reveal it as soon as possible. In fact reveal it the first thing - 'Are you okay with this ?. I can understand if you are not but make sure you do not bring this up in our future' - EXERCISE CAUTION. Some guys 'try' to be cool but they are not cool. You can see many answers here and I doubt if most of them can do it in practice. ONLY and ONLY if you are convinced that the guy is okay proceed for marriage.But do not repeat same mistake with this guy - if you do, you are complicating your life.There is still some probability for guy to say 'Okay. So you slept with this guy X some time back. But you are over it now. You realized that it was a mistake. That's okay. I am cool'. So it is better not to engage in premarital sex again with another person.Be it love marriage or arranged marriage, basing on the real life examples I have seen, most of the guys can not tolerate the fact that their girls had previous relationships. Even though they seem okay initially they keep bringing up comparisons later. Nothing can be more painful.Whoever disagrees with this are not looking at practical angle. My sincere advice is avoid sex until marriage. Sex changes the definition of the relationship. You can still be in love with a person without having sex. You can rationalize sex by terming it as 'love making' but it still changes relationship. It is like taking it to next level. It is safer for everyone if that next level is marriage. Even people who have premarital sex knew it is not right. (That is the reason they keep it away from parents). It can sometimes be dangerous. There are scenarios where the jerk videotaped entire thing and uploaded in the internet after dumping her. It might sound fancy to say 'I have an ex' or 'my ex did this' or 'while my ex and I were having sex...' on Quora. But real life ain't Quora.But you can not reverse what has happened. Be sensible while revealing such information.My longest answer on Quora - single word answer is. Yes he will marry. But it does not guarantee a happy marriageSomethings are best left unsaid. Besides what advantage are you getting by telling it to him? How is it going to make your present any better? Consider all the factors and take decision. Be practical.All the best.