Trigonometry

Cos(57 deg) = 0.5446 approx.

No. I is as described for the stated period.

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?

BEARING

1. A, B and C are three ships. The bearing of A from B is 045º. The bearing of C from A is 135º. If AB= 8km and AC= 6km, what is the bearing of B from C?

2. A helicopter pilot flies in the direction 147 degrees and lands when she is 12 km south of her starting point. How far did she fly?

Tanx can be written as tan0.5x by dividing it by 2.

tanx1/2=tan0.5x

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I doubt that you can, since the tangent of the whole angle is a function of the tangent of the half-angle and of the secant of the whole angle. Please see the link.

An obtuse angle.

you have to do the arcsin which is sin-1 on your calculator. i have not met anyone in my life who can do sin or arcsin in their head. not even my college teachers.

your theta is equal to 20degrees

If the angle is x radians, then tan(x) = 1/20 = 0.05

So x = tan-1(0.05) = 0.04996 radians = 2.86 degrees (approx)

nah

You use the arcsine or sin-1 function. For any value of sin(X), the function will return a value for the angle in the range [-pi/2, pi/2]. There are, however, infinitely many angles which will have the same value for sine. They are

X + 2k*pi and (2k+1)*pi - X radians where k is any integer.

If you still work with degrees, the answers are

X + 360k and (2k+1)*180 - X degrees.

It is infinite, in both directions. But it can be restricted to a smaller interval.

tan(3x)=1

3x= PI/4

x=PI/12 is the smallest positive number

10 * sine(9) = 1.5643

A stack of this height could be made with 1.000, .350, .114 and .1003 blocks for the minumum number of blocks.

26 = 64

There are 2 cups in a pint. There are 2 pints in a quart. There are 4 pints in a half gallon, like if you bought a half a gallon if milk at a store. There are 8 pints in a gallon.

10 pints equals 20 cups.

10 pints equals 5 quarts

10 pints equals 2 and a half gallons.

10 pints equals 1 and a quarter gallons.

Hope I helped!

When none of the angles are known, and using Pythagoras, the triangle is known not to be right angled.

cos(75 deg) = 0.2588, approx.

assuming that you mean what is theta if sin 4 theta = 0 then then theta=0, 0.25pi, 0.5pi, 0.75pi...

if not then without additional information the best answer you can get is sin4theta=sin4theta

1 knot = 1 nautocal mile per hour.

So 428 nm @ 20 knots per hour will take 428/20 = 21.4 hours.

After 12 hours, it is 20*12 = 240 nautical miles away.

Its distance East is 240*cos(1.4deg) = 239.9 nautical miles and

its distance South is 240*sin(1.4deg) = 5.9 nautical miles.

By the sine rule,

sin(C)/c = sin(B)/b

so sin(C) = 25/15*sin(32d15m) = 0.8894

so C = 62.8 deg or 117.2 deg.

Therefore, A = 180 - (B+C) = 85.0 deg or 30.5 deg

and then, using the sine rule again,

a/sin(A) = b/sin(B)

so a = sin(A)*b/sin(B) = 28 or a = 14.3

sin(30) = sin(90 - 60) = sin(90)*cos(60) - cos(90)*sin(60)

= 1*cos(60) - 0*sin(60) = cos(60).

-2(cot2theta)

The cosine of 10 degrees is 0.98480775301221. Hope I helped!

cosine = adjacent/hypotenuse