Special Angles
Special angles, or common angles, which have sines and cosines that are roots
of polynomials.
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Consider this figure.
Circle ABC has center D. AD, BD, and CD are radii. AC
is a diameter. BD ^ AC.
DE = EC, and that length is denoted a.
EF = BE, and that length is denoted x+a.
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We will set out to prove the following:
CD = 2a is the side of an inscribed regular hexagon.
DF = x is the side of an inscribed regular decagon.
BF = y is the side of an inscribed regular pentagon.
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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(36º) = 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 36º = 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 30º 60º 90º
triangle and the 45º 45º 90º 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 36º.
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(30º) = cos(60º) |
= |
sqrt(1/4) = 1/2 |
| sin(45º) = cos(45º) |
= |
sqrt(1/2) |
| sin(60º) = cos(30º) |
= |
sqrt(3/4) |
| sin(15º) = cos(75º) |
= |
sin 45º cos 30º - cos 45º sin 30º = sqrt(3/8)-sqrt(1/8) |
| sin(75º) = cos(15º) |
= |
sin 45º cos 30º + cos 45º sin 30º = sqrt(3/8)+sqrt(1/8) |
| sin(36º) = cos(54º) |
= |
sqrt(10-2sqrt(5))/4 = sqrt(5/8-sqrt(5)/8) |
| sin(54º) = cos(36º) |
= |
sqrt(1-sin² 36º) = sqrt(3/8+sqrt(5)/8) =
sqrt(5/16)+1/4 |
| sin(6º) = cos(84º) |
= |
sin 36º cos 30º - cos 36º sin 30º =
sqrt(15/32-sqrt(45)/32)-sqrt(5/64)-1/8 |
| sin(84º) = cos(6º) |
= |
sin 54º cos 30º + cos 54º sin 30º =
sqrt(15)/8+sqrt(3)/8+sqrt(5/32-sqrt(5)/32) |
| sin(18º) = cos(72º) |
= |
sin 54º cos 36º - cos 54º sin 36º =
sin² 54º - sin² 36º = (3/8+sqrt(5)/8)-(5/8-sqrt(5)/8) =
sqrt(5/16)-1/4
Note: this is exactly half the Golden
Ratio |
| sin(72º) = cos(18º) |
= |
sqrt(1-sin² 18º) = sqrt(1-5/16+sqrt(5)/8-1/16) =
sqrt(5/8+sqrt(5)/8) |
| sin(3º) = cos(87º) |
= |
sqrt((1-cos 6º)/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 75º cos 72º - cos 75º sin 72º = (sqrt(3/8)+sqrt(1/8))*(sqrt(5/16)-1/4)-(sqrt(3/8)-sqrt(1/8))*(sqrt(5/8+sqrt(5)/8))
= sqrt(15/128)+sqrt(5/128)-sqrt(3/128)-sqrt(1/128)-sqrt(3/8)*(sqrt(5/8+sqrt(5)/8))+sqrt(1/8)*(sqrt(5/8+sqrt(5)/8))
= sqrt(15/128)+sqrt(5/128)-sqrt(3/128)-sqrt(1/128)-sqrt(15/64+sqrt(45)/64)+sqrt(5/64+sqrt(5)/64) |
| sin(87º) = cos(3º) |
= |
sqrt((1+cos 6º)/2)
= sqrt(1/2+sqrt(3)/16+sqrt(15)/16+sqrt(5/128-sqrt(5)/128)) |
| sin(27º) = cos(63º) |
= |
sqrt((1-cos 54º)/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(63º) = cos(27º) |
= |
sqrt((1+cos 54º)/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(9º) = cos(81º) |
= |
sqrt((1-cos 18º)/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(81º) = cos(9º) |
= |
sqrt((1+cos 18º)/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
40º, because it's one of the roots of 8x3-6x+1, but, alas, it seems
that such an expression eludes me.
Internet References
http://hypertextbook.com/eworld/chords.shtml,
which cites Ptolemy's On the Size of Chords Inscribed in a Circle (2nd
Century AD).
Related Pages in this website
Common Angles -- a way for trig
students to remember the sines and cosines of the most common angles.
Trig Equivalences
Trig functions of special angles, part
2 -- cos 40º is one of the roots of 8x3-6x+1, so can we find
an arithmetic expression for cos 40º?
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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