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Special angles, or common angles, which have sines and cosines that are roots of polynomials.
We will set out to prove the following:
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| CF x DF + ED² = EF² | because | (2a+x)x + a² = (a+x)² |
| CF x DF + ED² = BE² | " | EF = BE |
| DB² + ED² = BE² | " | Pythagorean Theorem |
| CF x DF = DB² | " | DB² in the eq above takes the place of CF x DF in the eq above that |
| CF x DF = DC² | " | DB and DC are both radii, and equal |
| DF:DC is the golden ratio sqrt(5)-1 : 2 |
" | Look at triangle BDE. r=2a, so the sides are a and 2a, and the hypotenuse is x+a. (x+a)² = a²+(2a)² = 5a² x+a = sqrt(5)a, so x = (sqrt(5)-1)a, and since r=2a, we get... x:r = (sqrt(5)-1) : 2 |
| \DC = r is the
length of the side of an inscribed hexagon |
" | DC is a radius; r is the side of a hexagon |
| \DF = x is the
length of the side of an inscribed decagon |
" | Euclid, "the side of the hexagon and the side of a decagon
which are inscribed in the same circle... cut that line in the extreme and
mean ratio (Euclid XIII, 9)" (Ptolemy 20) a.k.a. the golden ratio. i.e. the ratio of x:r |
| \BF = y is the
length of the side of an inscribed pentagon |
" | Euclid again, "the square on the side of a pentagon is
equal to the square on the side of the hexagon together with the square on
the side of a decagon, all inscribed on the same circle. (Euclid XIII, 10)"
(Ptolemy 20) i.e. x²+r²=y² |
| sin(36�) = sqrt(10-2sqrt(5))/4 | " | Let r=2. Then y is the side of an inscribed pentagon. a=1, so x=sqrt(5)-1, so x²=6-2sqrt(5) y=sqrt(x²+r²)=sqrt(10-2sqrt(5)) Half the side of the pentagon divided by the radius is the sin of half the central angle, so sin 36� = y/4 = sqrt(10-2sqrt(5))/4 |
Here is a table of "exact" values of common values of sin and cos. I started with the standard values associated with the 30� 60� 90� triangle and the 45� 45� 90� triangle, and used the sum-of-sin, double angle and half angle identities to derive sines of other angles. Then I folded in Ptolemy's contribution, which is sin 36�.
In many of these expressions, we find sqrt(c+sqrt(d)) for two rational numbers, c and d. Often c+sqrt(d) is the perfect square of an expression of the form sqrt(a)+sqrt(b). Finding these is tricky. See Simplifying Nested Radicals for more info on this.
| sin(30�) = cos(60�) | = | sqrt(1/4) = 1/2 |
| sin(45�) = cos(45�) | = | sqrt(1/2) |
| sin(60�) = cos(30�) | = | sqrt(3/4) |
| sin(15�) = cos(75�) | = | sin 45� cos 30� - cos 45� sin 30� = sqrt(3/8)-sqrt(1/8) |
| sin(75�) = cos(15�) | = | sin 45� cos 30� + cos 45� sin 30� = sqrt(3/8)+sqrt(1/8) |
| sin(36�) = cos(54�) | = | sqrt(10-2sqrt(5))/4 = sqrt(5/8-sqrt(5)/8) |
| sin(54�) = cos(36�) | = | sqrt(1-sin² 36�) = sqrt(3/8+sqrt(5)/8) = sqrt(5/16)+1/4 |
| sin(6�) = cos(84�) | = | sin 36� cos 30� - cos 36� sin 30� = sqrt(15/32-sqrt(45)/32)-sqrt(5/64)-1/8 |
| sin(84�) = cos(6�) | = | sin 54� cos 30� + cos 54� sin 30� = sqrt(15)/8+sqrt(3)/8+sqrt(5/32-sqrt(5)/32) |
| sin(18�) = cos(72�) | = | sin 54� cos 36� - cos 54� sin 36� = sin² 54� - sin² 36� = (3/8+sqrt(5)/8)-(5/8-sqrt(5)/8) = sqrt(5/16)-1/4 Note: this is exactly half the Golden Ratio |
| sin(72�) = cos(18�) | = |
sqrt(1-sin² 18�) = sqrt(1-5/16+sqrt(5)/8-1/16) = sqrt(5/8+sqrt(5)/8) |
| sin(3�) = cos(87�) | = | sqrt((1-cos 6�)/2) = sqrt((1-(sqrt(9/32+sqrt(45)/32)+sqrt(5/32-sqrt(5)/32)))/2) = sqrt(1/2-sqrt(9/128+sqrt(45)/128)-sqrt(5/128-sqrt(5)/128)) = sqrt(1/2-sqrt(3)/16-sqrt(15)/16-sqrt(5/128-sqrt(5)/128)) or (this one has only two levels of sqrt nesting)... sin 75� cos 72� - cos 75� sin 72� =
(sqrt(3/8)+sqrt(1/8))*(sqrt(5/16)-1/4)-(sqrt(3/8)-sqrt(1/8))*(sqrt(5/8+sqrt(5)/8)) |
| sin(87�) = cos(3�) | = | sqrt((1+cos 6�)/2) = sqrt(1/2+sqrt(3)/16+sqrt(15)/16+sqrt(5/128-sqrt(5)/128)) |
| sin(27�) = cos(63�) | = | sqrt((1-cos 54�)/2) = sqrt(1/2-sqrt(5/32-sqrt(5)/32)) = sqrt(5/16+sqrt(5)/16)-sqrt(3/16-sqrt(5)/16) = sqrt(5/16+sqrt(5)/16)-sqrt(5/32)+sqrt(1/32) |
| sin(63�) = cos(27�) | = | sqrt((1+cos 54�)/2) = sqrt(1/2+sqrt(5/32-sqrt(5)/32)) = sqrt(5/16+sqrt(5)/16)+sqrt(3/16-sqrt(5)/16) = sqrt(5/16+sqrt(5)/16)+sqrt(5/32)-sqrt(1/32) |
| sin(9�) = cos(81�) | = | sqrt((1-cos 18�)/2) = sqrt(1/2-sqrt(5/32+sqrt(5)/32)) = sqrt(3/16+sqrt(5)/16)-sqrt(5/16-sqrt(5)/16) = sqrt(1/32)+sqrt(5/32)-sqrt(5/16-sqrt(5)/16) |
| sin(81�) = cos(9�) | = | sqrt((1+cos 18�)/2) = sqrt(1/2+sqrt(5/32+sqrt(5)/32)) = sqrt(3/16+sqrt(5)/16)+sqrt(5/16-sqrt(5)/16) = sqrt(1/32)+sqrt(5/32)+sqrt(5/16-sqrt(5)/16) |
In Trig functions of special angles, part 2 I had hoped to find an arithmetic expression that gives the cosine of 40�, because it's one of the roots of 8x3-6x+1, but, alas, it seems that such an expression eludes me.
| sin 0 = cos 90 = | 0 |
| sin 3 = cos 87 = | (1/16)*(-sqrt(2)-sqrt(6)+sqrt(10)+sqrt(30)+(2-2*sqrt(3))*(sqrt(5+sqrt(5)))) |
| sin 6 = cos 84 = | (1/8)*(-1-sqrt(5)+sqrt(30-6*sqrt(5))) |
| sin 9 = cos 81 = | (1/8)*(sqrt(2)+sqrt(10)-2*(sqrt(5-sqrt(5)))) |
| sin 12 = cos 78 = | (1/8)*(sqrt(3)-sqrt(15)+sqrt(10+2*sqrt(5))) |
| sin 15 = cos 75 = | (1/4)*(sqrt(6)-sqrt(2)) |
| sin 18 = cos 72 = | (1/4)*(-1+sqrt(5)) |
| sin 21 = cos 69 = | (1/16)*(sqrt(2)-sqrt(6)+sqrt(10)-sqrt(30)+(2+2*sqrt(3))*(sqrt(5-sqrt(5)))) |
| sin 24 = cos 66 = | (1/8)*(sqrt(3)+sqrt(15)-sqrt(10-2*sqrt(5))) |
| sin 27 = cos 63 = | (1/8)*(sqrt(2)-sqrt(10)+2*sqrt(5+sqrt(5))) |
| sin 30 = cos 60 = | 1/2 |
| sin 33 = cos 57 = | (1/16)*(-sqrt(2)-sqrt(6)+sqrt(10)+sqrt(30)-(2-2*sqrt(3))*(sqrt(5+sqrt(5)))) |
| sin 36 = cos 54 = | (1/4)*(sqrt(10-2*sqrt(5))) |
| sin 39 = cos 51 = | (1/16)*(sqrt(2)+sqrt(6)+sqrt(10)+sqrt(30)+(2-2*sqrt(3))*(sqrt(5-sqrt(5)))) |
| sin 42 = cos 48 = | (1/8)*(1-sqrt(5)+sqrt(30+6*sqrt(5))) |
| sin 45 = cos 45 = | (1/2)*sqrt(2) |
| sin 48 = cos 42 = | (1/8)*(-sqrt(3)+sqrt(15)+sqrt(10+2*sqrt(5))) |
| sin 51 = cos 39 = | (1/16)*(-sqrt(2)+sqrt(6)-sqrt(10)+sqrt(30)+(2+2*sqrt(3))*(sqrt(5-sqrt(5)))) |
| sin 54 = cos 36 = | (1/4)*(1+sqrt(5)) |
| sin 57 = cos 33 = | (1/16)*(-sqrt(2)+sqrt(6)+sqrt(10)-sqrt(30)+(2+2*sqrt(3))*(sqrt(5+sqrt(5)))) |
| sin 60 = cos 30 = | (1/2)*sqrt(3) |
| sin 63 = cos 27 = | (1/8)*(-sqrt(2)+sqrt(10)+2*sqrt(5+sqrt(5))) |
| sin 66 = cos 24 = | (1/8)*(1+sqrt(5)+sqrt(30-6*sqrt(5))) |
| sin 69 = cos 21 = | (1/16)*(sqrt(2)+sqrt(6)+sqrt(10)+sqrt(30)-(2-2*sqrt(3))*(sqrt(5-sqrt(5)))) |
| sin 72 = cos 18 = | (1/4)*(sqrt(10+2*sqrt(5))) |
| sin 75 = cos 15 = | (1/4)*(sqrt(6)+sqrt(2)) |
| sin 78 = cos 12 = | (1/8)*(-1+sqrt(5)+sqrt(30+6*sqrt(5))) |
| sin 81 = cos 9 = | (1/8)*(sqrt(2)+sqrt(10)+2*(sqrt(5-sqrt(5)))) |
| sin 84 = cos 6 = | (1/8)*(sqrt(3)+sqrt(15)+sqrt(10-2*sqrt(5))) |
| sin 87 = cos 3 = | (1/16)*(sqrt(2)-sqrt(6)-sqrt(10)+sqrt(30)+(2+2*sqrt(3))*(sqrt(5+sqrt(5)))) |
| sin 90 = cos 0 = | 1 |
http://hypertextbook.com/eworld/chords.shtml, which cites Ptolemy's On the Size of Chords Inscribed in a Circle (2nd Century AD).
Common Angles -- a way for trig students to remember the sines and cosines of the most common angles.
Trig functions of special angles, part 2 -- cos 40� is one of the roots of 8x3-6x+1, so can we find an arithmetic expression for cos 40�?
Sin or Cos 3x, 4x, etc. -- trig functions of any multiple of an angle.
Golden Ratio -- (sqrt(5)+1)/2, a special number that comes up in a variety of geometrical contexts
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