doing some trig and i came across this one problem that i just keep getting the same wrong answer so the question is
from a satellite 600 miles above the earth it is observed that that the angle formed by the vertical and the line of sight to the horizon is 60.276 use this information to find the radius of the earth.
this deals with right triangles
the way i’ve been trying to figure it out is
x/600= tan60.276
x=600 tan 60.276
which comes out to
1050
but in the book it says the answer is 3960 miles
b being the 600
angle A being 60.276 degrees
a being the earths radius
Your labeling is a little off. The line of sight to the horizon is tangent to the Earth, so in your figure B should be the satellite, angle B 60.276, C the horizon, and A the center of the earth. Then b=radius and c=radius+600. Then sin 60.276=r/(r+600)
I actually like calculus and LOGIC. Coming from a computer science background, clearly logic is one of the things I like the most. I really would love to just get a math degree for fun. Not to do anything career wise, but simply to have one.
My biggest problem is differential equations though. Man they’re tough. If I can get comfortable with that, I think I could be more solid at math.
It’s been awhile since I’ve been into mathematics very heavily though. i would love to get back on it but I’m always working and dicking around on message board. Need to tighten it UP!
I learned a lot from Coddington’s book. It’s largely self-contained and I don’t remember seeing the annoying phrase “the proof is beyond the scope of this book”. For $8 you really can’t lose. The Tenenbaum book in the related items also has good reviews but I’ve never seen it.
If you want to learn differential equations, I know a place where DEs are regularly used in practical applications, and I think practical applications help motivate the learning process: A Control Systems perspective.
There are many, MANY other fields that use DE’s, but I find Control Systems to be fun because today it can be, how do I regulate molten steel such that I get sheets of steel out the other end of my machine? How do I use an electric motor to control an inverted pendulum? It can be mechanical or digital. You can work it out on paper, run it through a simulation on a computer, or you can really implement it. You also have to understand the physical system well, which is fun if you like figuring out physics.
yeah, with this identities in my opinion is best to go from both sides and reach a common point because the steps (random identities, factorizations) are not always easy to think about.
wow i’m still not getting it i think i’ll have to go visit my teacher during office hours i greatly appreciate all of you trying to help.
everything before trig identities has been a breeze.
I’ll let t=tan x , s=sin x , c=cos x to save space. Try it your way, but you can’t multiply by just t-s, you have to mulitply by (t-s)/(t-s)=1, so you don’t change the left hand side. Then it might be a good idea to write everything in terms of s and c, by substituting t=s/c everywhere.
Warrior’s Dreams’ method is easier though. Just cross-multiply, since A/B=C/D if and only if AD=BC. Once you do about 10 (with the help of a sheet of trig identities ideally), they get really easy, although somewhat boring.
**Know your algebra. **It sounds obvious but in my experience and from what I’ve heard from others, weak algebra skills are a huge problem. This includes exponents/logs and you should know some trig too. If you know algebra well, you’re already ahead of a lot of people.
The main ideas are pretty easy to understand, especially if you have a physical interpretation. For example, the derivative of a function is the slope of the tangent line and also the velocity of a moving car (if you interpret the x value as time and y as distance from some point). It’s easier to grasp the concept with the concrete model of the car than the seemingly unimportant and unmotivated problem of a tangent line. I’m not saying the tangent line interpretation is wrong or somehow worse, but physical examples provide good motivation and intuition.
Look at The Fundamental Theorem of Calculus, or at least one half of it. It probably looks pretty intimidating, real genius level math lol. It took geniuses to discover it, but anybody can understand it. It just says that if you know the velocity of a moving car during some time interval, then you can calculate the change in its position during that interval. It actually says more because it gives a way of computing the change in a way that seems completely obvious. **Physical models and geometric intuition are important, **not just for calc 1 but for all math. Just to use an example from more advanced calculus (multivariable calc, so calc 3 I think), I can’t imagine anyone having a clear idea of torsion without having some kind of geometric picture in their head. That animation on the side is worth more than the equations next to it. It doesn’t matter how good a mathematician you are, the equations only really mean something if you know the idea behind them.
To kind of expand on what people have been saying, the thing you should watch our for is is to make sure you can do PreCalc things like it’s second nature. I will assume you have Stewart’s textbook, but if not, you need to be able to look in the reference pages and be able to say “I can apply this.”
Everything you (should have) learned in algebra, geometry, can and will be used in Calc.
Algebra:
By now, performing elementary arithmetic operations should be second nature. If not, you’re in trouble. You should know at this point how to add and multiply fractons and exponents, as well as simplifying algebraic expressions. You will be expected to show your work on quizzes/exams and not being able to simplify something like (x(x+y))/x^2 is a red flag before you even start the class. Why?
You will get certain kinds of limit problems that you can solve by “plugging” the limiting value of x into the function, but only after you simplify the expression.
For instance
the limit of (x-1)/(x^2 -1) as x goes to 1.
As written, you can’t evaluate it directly, but using the identity x^2 - 1 = (x+1)(x-1), you can rewrite the denominator and the (x-1)'s cancel out to where you now have the limit of 1/(x+1) as x goes to 1, which is 1/2. Replacing the original function with this one is valid because the domain of both functions are the same except for when x = 1, but in calculus we only care about what happens around x = 1 and not AT x = 1.
Oh yes, domain and range should be familiar terms to you. If not you need to review them, as well as what it means to be a function. The absolute value function is important to know and understand how to use.
Also, you will be expected to have a small inventory of “basic” functions under your belt such as x, x^2, x^3, 1/x, 1/x^2, all 6 trig functions, e^x, and ln(x) to name a few. You will be asked on a quiz/exam to plot these functions and look for things like where they intersect (do you understand how to find such points?). You will also be expected to know how to “translate” functions, that is, take functions that I have mentioned and “move them” to different parts of the graph. For instance, if I take x^2 and want to shift if c units to the right, then my new function is (x-c)^2. You will also be expected to know how to reflect elementary functions about the x- and y-axes (which involves multiplying the original function (or its domain) by -1).
It goes without saying, but the equation of a line, and how to find it, should also be straightforward. Not knowing that is a red flag as well, mainly because you will be looking for tangent lines throughout the entire course.
Geometry:
Although it is commonly referred to as “Calculus,” there is quite a bit of geometry to it. You find tangent lines to curves, you use the limiting concept to find areas under a curve through rectangles. When you get to related rates problems, you will need to know things about cones, spheres, cylinders, etc. You may also do hyperbolas, parabolas, and ellipses as well and should be familiar with those. Some problems require the use of the Pythagorean theorem. Others don’t even require calculus at all. One common example is the following: you are asked to find the area under a boxed up region, which turns out to be in the shape of a triangle which has two of its sides on the x and y axes. So you just use the area of a triangle formula to find it. You can use calculus, but it’s kind of inefficient.
Trig
Think you’ve seen the last of trig? Think again. Each trig function has its own derivative. You’ll see these a lot when you will do polar coordinates (which may be second semester) and especially in solving particular kinds of integrals. These especially you need to know how to use trig identities to simplify expressions to where you can actually integrate it. You’ll also need to know the unit circle because it’s inconvenient to have to look up sin(7pi/6) if you don’t have a calculator on you, so you may as well just memorize them.
Well, that’s just a small sample of what you should know and how it will be applied. There’s tons of stuff I left out like curve sketching, using derivatives to find out info about the shape of a graph, and the relation between integrals and derivatives, but this is honestly what you should have a handle on by now math-wise. Otherwise, this next semester will be difficult for you. If you’re freaking out, you should be. Better you find out about this now and give yourself time to prepare than later on when it becomes harder to catch up. This stuff will snowball on itself pretty quickly too, so knowing all this stuff before you step into the classroom will give you an advantage over most people.
Speaking of calc 1, I just saw this today. It’s pretty much spot-on about the uselessness of formula memorization and mechanically doing things you don’t understand. I’ve also wondered why things like the product rule aren’t proved in class more often since literally all it comes down to is adding zero (which, along with multiplying by 1, is an extremely useful trick that comes up often). Then again, with generally weak teachers teaching only to the test, I guess it’s obvious why.
Timothy Gowers is a Fields Medal winner, so he knows what he’s talking about. It’s unfortunate that this experience was surprising to him though. Respected mathematicians need to speak up more about the state of math education, but most of them seem unconcerned and even too timid to take action. Then of course they complain that their freshmen don’t know math! At least this guy is trying, and Lockhart definitely isn’t timid.
That would require critical thought in a class that is essentially a watered down version of analysis. Most people in these classes can’t even argue using definitions, or do a proof by contradiction.
