General

Here's an anecdote. So I am currently on an internship, which is about to end. The company is looking for new candidiates, all from the same school. Among the ones they selected, they weren't the ones with the highest GPA or the ones with their 50 different types of extracurricular. The ones chosen for interviews were the ones that had more experience in the relative field. So at the end of the day, you could come from an average college, but if you have good experience in your field, you are more likely to be chosen for an employment based on that rather than where you went to for college.

You're not screwed. If you're fine in everything else, life's good. Here's a somewhat inspiring story.

Trashed myself in HS, did nothing but sleep and play Maple, don't know how I graduated. By the greatest fluke I was accepted into a non-selective college but has a hidden, unforgiving science/biology program. Stupidest thing I've ever done was choose to be a bio major going into pre-med. Promised myself if I didn't graduate top of my class, I'd end my life on graduation day because I'd feel too ashamed to live to see another day. How's that for commitment?

So here's what the past four years of my life have looked like.

--

Freshman Year:
Class begins at 8, doesn't get out til 5 every day, 5 days a week. First semester, 16 credits. Worked my ass off, 4.0. Same story for second semester.

Sophomore Year:
First semester - Took three junior/senior classes in my first semester, got As in everything but one. Second semester - repeat of previous semester. Slept in class, studied outside of class where there was nobody to bother me. Not the best year, but not the worst.

Junior year:
First semester - Took senior course as a junior. To illustrate how I studied, here were my habits: 4 hours of sleep, get up, drive to school, study for 3 hours, take a 20 minute nap. Wake up, pop 8 caffeine pills, down two 5-hour energies, take a double shot of espresso. Stumble down to class, willing to die before I get a B on an exam. Walk out of the exam, pass out in the hallway, get 100%. Rinse and repeat. Second semester - took senior/grad level courses. I spent the entire five months sleeping underneath my desk so I would have to wake up every thirty minutes or so. Got ~2-3 hours of sleep the week before exams. Only student to get an A in the class. Applying for Med schools, gets interviewed by six profs who write combined rec letter for schools; receives highest ratings.

Senior Year:
First semester - finished up all the courses I was supposed to take as a jr/sr, got nothing but req courses to finish up on. Thought I could take it easy. Not happening.

Second semester - Advisor tells me last minute I need two more credits to graduate; packs in 18 credits/semester, has two part time jobs, took night classes totaling 50 hrs/wk on school, outside class, and work. Woke up at 6AM every morning, didn't get back until 11PM at night. No free weekends. Forms of nourishment were limited to Honey Nut Cheerios, Ramen, and bottled water. Hourly headaches worse than any hangover. Not sure how I'm still alive.

But it all pays off: Graduating next week, Magma with Honors.

[b]TLDR: Moral of the story - [/b]It only gets harder but at least you can feel proud of yourself. Somewhat.

@glyph: I see; I'm guessing you're having issues with manipulating some of the identities or perhaps algebraically getting to the point to where you can start substituting said identities into some-odd variable.

In terms of the unit circle (and perhaps this is a bit of a convolution on my part), I usually like to visualize (assuming you are talking about algebraic identity manipulations; correct me if I'm wrong on this, I'm also assuming that you can work around derivatives and integrals just fine) how they relate to each other on a unit circle.

Let's use something relatively basic to demonstrate this point:

Using the sum/difference formula of cos(a-b)=cos(a)cos(b)+sin(a)sin(b) in terms of the already-solved problem of:

cos(15degrees): (here is the algebraic solution so we're clear on what is being talked about:: http://imgur.com/CllV8XY
On a circle who's radius is r=1 (essentially the unit circle) we can geometrically represent the same problem as:: http://imgur.com/g0JK25t

(excuse the sloppiness of the drawings, I have crappy dry-erase markers at the moment)

-note that the area where the solution would be found is in the intersection between the two special triangles

Without going too much into what the algebra of the mentioned sum/difference formula actually means, that's essentially how I think of a lot of trigonometric equations and identities (you can pretty much do what I just did with any identity or formula in trigonometry; some taking longer than others).

Now, algebraically manipulating them in terms of things like integrals and derivatives starts to get a bit more complex, but just using your basic special triangles on a r=1 circle with base sinusoidal waves in mind, you could also visualize what exactly these integrals and derivatives are really doing. Personally, I would strongly try messing around with them yourself sometime; it really helps internalize the mathematics. However, also keep in mind that algebra is simply the language of patterns and that a lot of the manipulation you are doing is really just mathematical induction with other aspects of the patterns in mind. Thus, not all manipulations can readily be visually represented (at least not without the proper software).

Let me know if I missed the point of your question; I haven't taken to much answering of math questions on the internet in the past.

Literally had to cram learning an entire semester of metabolic pathways and regulatory networks into three days for my biochem final that was on the first day of finals. So. Many. Mechanisms.

Then I had to do a 20 page take home final for my quantum chemistry class, a 10 page paper, and study for another final over the course of three days. Somehow managed score a job in the middle of doing all of that too.

Just saying, where you go for undergrad doesn't really matter. I'd honestly say it's almost better to go to a worse state school where you can easily make good grades than a higher tier university where the classes will be much harder. As long as you make good grades in college and take advantage of the resources your university offers, you can probably get into a decent grad school. Or med school or whatever you think you want to do after you're done with college.

@sezbeth: I mean more something like

Gm^2 = sec(x)*F*g(sin(x) - 4Lsin^2(x) + 4l^2sin^3(x)), where x is the only unknown quantity, and I would like to solve for it in terms of the other quantities.

Then the algebraic manipulation gets trickier for me, though I'd be very happy if you could tell me of a way to think about this on the unit circle.

The example you gave is no problem for me at all.

@juliacat Don't worry, you still have hope.

But here is one thing to consider: if you are failing AP trig in highschool, there's a good chance you are going to struggle with similar courses in college. So even if you get into an ivy league, imagine what would happen once you're in.

I think the problem here isn't whether you can get into an ivy league, but rather your work habit. How are you studying that you are failing trig?

@glyph: This depends; what are you interested in knowing? If it's the algebraic equations, then what specifically is giving you a hard time? Things like 2sin(pi/2+3pi/4)=2 ? If you could, find a visual example and I'm sure I could break it down fairly easily.

The interesting thing about trigonometry is that despite the later algebra, all of it can still be proven on a basic unit circle (and you don't even have to memorize that).

I apologize if I'm coming off a bit strong with the questions, but I want to know exactly what I'll be explaining before going into a two-page long explanation about how unnecessary half of the memorizing in trigonometry actually is.

@sezbeth: Please do tell.

Trig was extremely difficult for me until I was further introduced to the complex plane. I'm still not quite accustomed to it, but I find normal algebraic manipulation of trig functions much harder than the more geometric approach.

It's a lot to ask someone of your age where in 5-years/10-years do they see themselves or what in that same time are they doing? Unfortunately, our own overbearing parents just give flat out bad advice. I have some thoughts to share on what happens AFTER college to hopefully get you thinking about what you could be doing.

I believe that the first university you attend does NOT have to be highly prestigious.

If your outlook is to work after graduating college with a bachelors, then admission into a highly prestigious school like an Ivy League is overrated. It's also expensive and competitive, which will probably hurt your future. Honestly, you'll be no worse with a in-demand degree earned from a reputable state university/college. You'll probably have a better GPA to talk about (if potential employers even ask) and less debt to pay off. Due to the high cost of living in some cities, being able to work as soon as possible with preferably a midrange salary is on the radar for most recent grads. Once you have a good work history on your resume, your education becomes way less important.

Even if you intend to continue your education by attending graduate school, going to a less prestigious, yet still reputable university after high school is still a very good option. Much of what I wrote in the preceding paragraph applies here too. Your grad school application will look more attractive if you have better grades, relatable activities, and connections (for recommendations.) You might be able to get admitted to a prestigious grad school.

So yes, attending a highly prestigious institution for your undergraduate years doesn't mean much in the real world. Employers really don't care. Instead, they're looking for qualified, or even better, well connected applicants (unless you're in engineering.) When you write a resume, your education history is going to take maybe at most three lines out of probably 50. The name of the school is not that important, even probably less important than the degree major on the line below.

Things have changed in the working world. As an example, around where I live there are lots of new hipster creameries and ice cream parlors. Five years ago it was tea bars. Restaurant and hospitality is not a joke field. Our parents said a college degree was the path to not flipping burgers, but these young small restaurant owners are pulling in huge profit margins churning ice cream and blending drinks in-house. A $4 8 oz smoothie or $5 cup of ice cream is a bad buy, but I'm with my friends being cool. Anyways, the change that made this happen was Yelp and Facebook, both which became big in 2009. Those platforms gave small businesses a way to explosively publicize themselves amongst other young people using the internet. If you have never been to one of these hip desert places, then I will tell you that one of the newest creameries around here has a line with a wait time of 2 hours on a Saturday night. That's very good business for a small group of new college grads working together. Several college degrees target this field - business, marketing, communications, and hospitality. We are also seeing many young people in the face of the ongoing weakened economy, reinvent traditional marketing strategies using their own creativity and hard work. Publicity attracts new customers and if you have a good product, you will build a loyal base. It's just we now have the internet as a growing medium for advertising.

It's unfortunate that some people, such as our parents, have dreamingly high hopes that their children will enter highly prestigious professions such as medicine and law. Undergrad biology degrees are too commonplace and thus devalued. Similarly, undergrad political science and economics degrees are not that useful as they're also kind of designed to be stepping stones to more education. Worse yet, higher education is the most expensive it's ever been and graduate school is not financially sensible or even flat out not affordable for many people especially after paying out heavily for their undergraduate degree.

I find that a lot of students tend to vastly overthink trigonometry; in truth, you don't have to memorize as much as a lot of high school curricula will teach you. If all of the unnecessary memorization is what is bothering you, I'm willing to toss out a few interesting things that may help you out.

trig is stupid so it doesn't matter if you fail

Well, I'm attending a pretty low ranking college, but I've already secured a co-op for this summer as a freshman and most likely looking forward to an internship next summer. So even if you don't make it into the Ivy Leagues, you can still make something of your experience.

Remember, it's not the college that you go to; it's the opportunities that you seize.