Can you explain why sin(1+phi) is so near to 0.5?
85243
1+phi is almost 2.618... 2.618 divided by pi is 0.833...\nso, 2.618 in radiants is nearly 8/10 of pi that is 4/5 of pi, that is 4/5 of 180° --> 144°; it's sufficiently near to 150°, whose sin is 0.5 being equal to 5/6 of pi.
908924
Because of (one of) the definition(s) of sin and cos.\nWhich is: take a unit circle, and draw in any radius at some angle theta. Then sin theta = the y value, and cos theta = the x value. Those are obviously less than 1.\n\nOr you could just take a typical right angled triangle - the length of the opposite/adjacent sides are always less than the length of the hypotenuse.
763980
Look at the decimal expansion of phi (the golden ratio) and you'll have your answer...\n\nФ = 1.618033988749894...\n\n1, 6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 4, 9, 8, 9, 4, 8, 4, 8, 2, 0, 4, 5, 8, 6, 8, 3, 4, 3, 6, 5, 6, 3, 8, 1, 1, 7, 7, 2, 0, 3, 0, 9, 1, 7, 9, 8, 0, 5, 7, 6, 2, 8, 6, 2, 1, 3, 5, 4, 4, 8, 6, 2, 2, 7, 0, 5, 2, 6, 0, 4, 6, 2, 8, 1, 8, 9, 0, 2, 4, 4, 9, 7, 0, 7, 2, 0, 7, 2, 0, 4, 1, 8, 9, 3, 9, 1, 1, 3, 7, 4, 8, 4, 7, 5...\n\nLook through until you find 16 consecutive digits that add up to 90.  \n\nThe first set is 4989484820458683 = 90\nThe next set is 9894848204586834 = 90\nThe third set is 8475408807538689 = 90\nThe fourth set is 4494704956584678 = 90\n\nLooking at the expansion I figured out the fifth set:\nThe fifth set is 5884607499887124 = 90
981364
Gopal has the correct answer! \n\nSo... becareful with some other results... and... never use:\n0.833 instead of 5/6... it's a mistake!\n\nFinished Algebra with 19!
257603
Sin A = -12/13\nCos A = 5/13\n\nFirst, draw a x-axis and y-axix. The angle starts on the positive x-axis going counter-clockwise ends at the line that contains (0,0) and (5, -12). That line is in the fourth quadrant so angle A is bigger than 270 degrees but less than 360. Angle A's sin and cos are the same as the angle that starts on the positive x-axis and goes CLOCKWISE to the that line. Let that angle be angle B.\n\nDraw a line from (5, -12) up vertically to (5, 0) and a line from (0, 0) to (5, -12). You get a triangle with Angle B. It's opposite side is 12 units long, adjacent is 5, and hypotenuse is 13. \n\nTherefore the sin is 12/13 and the cos is 5/13. Now we look at the position of angle B. Because angle B is in the fourth quadrant, its sin must be negative and its cos positive.\n\nWe get sin B= -12/13 and cos B= 5/13. And since angle A's sin and cos are the same as angle B's...\n\n:)
417496
It is divergant. Because the limit of the sentance to infinity is 1/2 and not zero. ( It should be zero so that the summation converges)
22472
It's well known that lim x->0 sin(x)/x=1\nThen lim x->inf. sin(1/x)/(1/x)=1\nwhich is exactly what you need.
925769
sin(θ) ≈ θ\nSo cos (θ) ≈ 1  because cos (0) = 1\nMore accurately cos (θ) ≈ 1 - θ²/2
1159205
pair up the  opposite sines, for example sin(10) and sin(350) and add them:\nsin(10) + sin(350) = 0\nsin(20) + sin(340) = 0\n...\nuntil you only have sin(180) and sin(360) which both equal 0\n\nPS. I assumed you were using degrees (sexagesimal). Won't work in radians (RAD) or metric (decimal) angles.
464358
davidosterberg1 is correct\n\nbelow is a link to the explanation\n\nhttp://id.mind.net/~zona/mmts/trigonometryRealms/radianDemo1/RadianDemo1.html
730722
put sinx=t\ncosxdx=dt\nI=fcos(pi t)dt with limits -1,+1\nI=(sin(pi t))/pi with limits +1,-1\nI=(sin(pi)-sin(-pi))/pi=0\nthanks & good luck
911476
5sin² x +13sin x - 6 = 0 This is a quadratic in sinx. Solve by factoring (if possible)\n\n(5sinx - 2)(sin x + 3) = 0\n\nOnly possibility is sin x = 0.4 as |sinx| ≤ 1 in real numbers. So sin x = -3 is disallowed.\n\nThus x ≈ 0.4115, π - 0.4115\n≈ 0.4115, 2.730
846932
The x value will be (sqrt(5) - 1) / 2, or -.618, which is 1 - phi = -1 / phi, where phi is the golden ratio.  If f_n is a fibonacci number, the ratio of f_n/f_(n+1) approaches phi - 1 as n approaches infinity.  Since the terms of your sequence are of the form f_n + f_(n+1)*x, they would approach f_n + f_(n+1)*(-f_n/f_(n+1)) = f_n - f_n = 0, a convergence.
611271
a_i = sin(phi_i)\nsin^3(phi_i) = (3sin(phi_i) - sin(3phi_i)) / 4\nso: 0 = Sum[1..n] sin^3(phi_i) = \nSum[1..n] (3sin(phi_i) - sin(3phi_i)) / 4\nso: Sum[1..n] sin(phi_i) = 1/3 Sum[1..n] sin(3phi_i) <= n/3\nqed
1030442
The Golden Ratio is approximately 1 to Phi (approx 1.618) or 0.618 to 1 (mathematically both are the same). Basically it's the ratio you get when you a divide a line AC such that AB/BC = BC/AC.\n\nAre you familiar with the Fibonacci sequence? Each number is the sum of the two preceding numbers:\n1 + 1 = 2\n2 + 1 = 3\n3 + 2 = 5\n5 + 3 = 8\n8 + 5 = 13\nand so on. So the sequence runs: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. \n\nIf you take any number in the sequence and divide by the preceding number you get a number close to Phi (approx 1.618).\n89/55, for example, is 1.6181818181818...\n\nThe higher you go up the sequence, the closer you get to an accurate Golden Ratio.\n\nThe ratio appears frequently in nature (for example, the ratio of your forearm to your upper arm is about 1 : 1.618), and because of it's aesthetic appeal it has also used frequently artists and architects throughout history. The pyramids and the parthenon use the Golden Ratio.
1186234
Actually, when the answers come out as negative, it's calculating in radians. So the sine of 6 is negative, as it is actually calculating about 324 degrees.\n\n2nd+0 resets your calculator, but i don't recommend it.
626033
I wanted to give a partial answer that wasn't presented elsewhere:\n\ne.g. Sine 10 deg =0.174, while cosine 80 deg= 0.174. how ironic is that?!!\n\nIt's not ironic at all.  If you have a right triangle, you know one angle is 90°, and the sum of all angles must be 180°, so the other two angles must add to 90.  You give sin 10°, so the other angle must be 80°.\nThe sin of any angle must be equal to the cos of its complementary angle (x and 90°-x are complementary), because of the definitions.  The side opposite the x angle is adjacent to the the 90°-x angle.\nThus sin 10° = cos 80°.