Trigonometry

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

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

sin-1 (0.91) = about 1.14328 radians.

(40000000 x .00002)=800

the answer is (area)

If you have the sine of an angle and you need to find the angle, you have a

few choices. They're exactly the same choices that you have if you're going

the other way ... you have the angle and you need to find its sine:

-- pocket calculator, or computer . . . the way it's most often done these days

-- look it up in a table in a reference book

-- calculate it on paper from an infinite series . . . technically possible, but no reason

to subject yourself to the heartburn unless you need extreme precision, better than

what you can get with a 10-digit calculator.

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?

Sec(2x) = 1/Cos(2x)

Putting a question mark at the end of a phrase does not make it a sensible of even an answerable question.

Sine and cosine of real numbers? Yes, they do exist. In fact, sines and cosines of complex numbers also exist. Does that answer the question?

Not enough information has been given to solve this problem such as: What is the angle of elevation?

Name the rectangle ABCD,letting the larger side be AB(7cm) and the smaller side be AC(5cm). Draw the two diagonals across the rectangle. Name the point of intersection of the two diagonals (which resembles an X) F. Name the midpoint between the smaller side AC, to be E.

AFC is the acute angle you are looking for, angle AFE is half of that.

So AFE is a right angled triangle and AE is equal to half of 5cm, which is 2.5; FE is equal to half of 7cm, which is 3.5.

Now, taking triangle AFE,

Tan AFE=2.5/3.5

AFE=35.537 degrees

Since the angle AFE is half of the acute angle AFC that you are looking for,

multiply AFE into two

So, 35.537*2

=71.075 degrees

=71.1 degrees (to 1 decimal place)

AFC= 71.1 (ANS)

y = -1 + 3 sin 4x

Let's look at the equation of y = 3 sin 4x, which is of the form y = A sin Bx, where

the amplitude = |A|, and the period = (2pi)/B.

So that the amplitude of the graph of y = 3 sin 4x is |3| = 3, which tell us that the maximum value of y is 3 and the minimum value is -3, and the period is (2pi)/4 = pi/2, which tell us that each cycle is completed in pi/2 radians.

The graph of y = -1 + 3 sin 4x has the same amplitude and period as y = 3 sin 4x, and translates the graph of y = 3 sin 4x one unit down, so that the maximum value of y becomes 2 and the minimum value becomes -4.

Sin theta of 30 degrees is

1/2

They are the projections, onto the x and y [Cartesian] axes, of a point whose polar coordinates are (R, theta).

It's a common Trig way to express a point when a radius is rotated around a given angle. For example, where exactly would the edge of a two foot gate lie if the gate opened 30 degrees? R is two feet. Two times cosine 30 is the x coordinate and two times sine 30 is the y coordinate.

at a 45 degree angle, or pi/4

The area is 99.0 square units.

Provided that any denominator is non-zero,

sin = sqrt(1 - cos^2)tan = sqrt(1 - cos^2)/cos

sec = 1/cos

cosec = 1/sqrt(1 - cos^2)

cot = cos/sqrt(1 - cos^2)

22227 feet

This is a right triangle with an angle equal to 13 degrees, and the side opposite that angle equal to 5000 ft (6000-1000). We need to find the hypotenuse (which we will call x) of the triangle (i.e., the length of the trail). Because sin(theta)=opposite side/hypotenuse.

Therefore we have:

sin(13 deg)=5000/x

Solving for x:

x=5000/sin(13 deg)=22227

[Be sure that you are using degrees rather than radians; otherwise, you will get a wrong answer. If your calculator is using radians and you don't want to change it, you first need to convert 13 degrees to radians by multiplying 13 by (pi/180).] Using radians, you would do the same as above, but the last line would be:

x=5000/sin(13*pi/180)=22227

SOHCAHTOA
This would be the sine, the opposite over the hypotenuse.
5000/h=sin 30 degrees
5000 divided by sin 30 degrees= trail

No more than means the number or less.

For example, no more than five could be 1, 2, 3, 4, or 5 but no higher. It couldn't be 6, 7 or 8 etc.

that's right, but to added on -1, -2, -3, -4, or -5 can be too.

Loads and loads of formulas.

That's trigonometry. ;-)

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