Trigonometry

The period is 1 millisecond.

sin(3x) = 3sin(x) - 4sin^3(x)

Since secant theta is the same as 1 / cosine theta, the answer is any values for which cosine theta is zero, for example, pi/2.

This would be pi/4 radians or 45 degrees.

192 = 361

sine(sin) = opp/hypcosecant(q) = hyp/oppcosine(cos) = adj/hypsecant(q) = hyp/adjtangent(tan) = opp/adjcotangent(q) = adj/opp

sin(62 deg) = 0.8829

Probably you should start by looking up the double-angle formulas, reducing the "4a" to some combination of "2a".

It depends. Between 9th to 12th.

sin(70 deg) = 0.9397 approx.

it can be written as follows

nf=m+a=R

Where n stands for needs, f for faith,m for motivation, a for action and R represents result.

Secant is 1 over cosine and cosine 0 equals 1.

Simple harmonic motion (SHM( is defined by the second order differential equation:

d2y/dt2 = -ky

where y is a fubction of time, t and is the displacement (relative to the central position), and k is a positive constant.

The equation says is that at any given position of the object undergoing SHM, its acceleration is proportional to its displacement from, and directed towards the central position.

The sine and cosine functions are solutions to the differential equation.

The tangent of 0.47 radians is about 0.508. The tangent of 0.47 degrees is about 0.00820.

By converting everything to sines and cosines. Since tan x = sin x / cos x, in the cotangent, which is the reciprocal of the tangent: cot x = cos x / sin x. You can replace any other variable (like thetha) for the angle.

Cos(57 deg) = 0.5446 approx.

The formula for converting radians to degrees is the given angle in radians multiplied by (180 degrees / pi), so in this case, 7 pi radians would be equal to 1260 degrees. (By the way, the greek letter pi isn't spelled with an e).

I've read a different version of the 1729 story, in which someone was visiting him and they remarked on the unremarkable quality of the number, which Ramanujan contradicted with the interpretation you give [roughly]. There's certainly a chance he had played with such numbers earlier and remembered previous reasoning, rather than coming up with it on the spot. Even the story I read is a bit apocryphal; it should really have a reliable citation.

I also remember reading a slightly different version of your house number problem, where Ramanujan says that he immediately knew the solution must be a continued fraction, and solved the whole class of problems. That version didn't actually describe the problem, and I have trouble following your description; sorry.

It's probable that Ramanujan only wrote his conclusions in his notebooks, having worked the results out elsewhere. I'm interested in the 5 formulas you say haven't been proven: could you list them?

I find that not having access to a computer forces ingenuity in performing calculations. For instance, in the story about Gauss summing the numbers from 1 to 100, if he had access to a scripting language he might have just had a computer find the answer instead of figuring out the triangular number formula. In any case, perhaps the lack of access to computers helped Ramanujan in some way.

I don't believe your assertion that he never showed any proof of his work. I wish I had access to some of his published work to see for myself.

Ramanujan was very rare and had an amazing intuition. Comparing him to current university math students is like comparing them to Gauss--it's an unfair comparison.

How would math be different if he had survived for longer? It's completely unclear. He probably would have continued making numerous deep and surprising discoveries related to analytic number theory. Perhaps something would have been groundbreaking, or perhaps they would just be interesting curiosities. Who can say?

C(p) = 0.59p

No. I is as described for the stated period.

Tangent(62 degrees) = 1.88073 (rounded)

Tangent(62 radians) = -1.09751 (rounded)

The value is 0.

S', the complement of a set S, in the context of the universal set U, is the set of all elements of U that are not in S.

It is important to note that a complement is defined only in terms of the universal set. The following, rather crude example illustrates the point. Suppose S is the set of all boys.

Then S' may be the set of all girls if U = youngsters;

or S' = set of all girls, women and men if U = people;

or S' = set of all girls, women, men, dogs, cats, cows, ... if U = mammals; and so on.

As you see, changing U alters S'.

Without an equality sign it's not an equation but it can be simplified to: 7x-5

This isn't hard to figure out if you know how far you want the base of the ladder from the wall. All you have to do is use the Pythagorean Theorem. The theorem is A2 + B2 = C2. A and B are the short sides and C is the hypotenuse (the side opposite the right angle). When dealing with a ladder, the ladder is the hypotenuse, or the C in the equation. Another way to put the formula is to multiply A by itself and B by itself, add them together, then take the square root of the result.

Decide how far you want the base of the ladder from the building. Multiply that by itself then add that to 169 (that is the square of 13). Then take the square root of that and you get the ideal ladder height. Lets assume 5 feet from the wall. The square of that is 25. 169 plus 25 gives 194. The square root of 194 is 13.928, and we would round up to 14 feet.

If you mean a step ladder, and you are trying to reach a ceiling, then you would need roughly the height of the room minus your own height. You might want to add a tad more. You could make it shorter, but that would not be wise, since you should always avoid the top two steps if possible.