Sine of 36 DegreesDate: 11/18/2001 at 21:07:38 From: Steven Kilby Subject: Trig In Methods of Computing Trig Functions http://mathforum.org/dr.math/problems/morgan.9.07.99.html Dr. Rick wrote that Ptolemy geometrically calculated the sine of 36 degrees using the construction of a regular pentagon. How would this work? Date: 11/19/2001 at 16:11:22 From: Doctor Rick Subject: Re: Trig Hi, Steven. Draw a regular pentagon. Its vertex angles are 180*3/5 = 108. Draw the diagonals. The triangle formed by one diagonal and two sides of the pentagon has angles of 108, 36 and 36 degrees. The triangle formed by one side and parts of the two closest diagonals has the same angles, so it is similar to the other triangle. The larger triangle is also divided into two of the smaller triangles and another isosceles triangle. Call the sides of the pentagon x, and the equal sides of the smaller isosceles triangle y. Thus: B /\ x / / \ \ x / /y y\ \ / / \ \ __________/________\___________ A y D E y C You can show that triangle BCD is isosceles, so that DC = x. Then the similarity of ABC and ADB gives this proportion: x/y = (x+y)/x Rearrange to get a quadratic equation in y: x^2 = y(x+y) x^2 = xy + y^2 y^2 + xy - x^2 = 0 Using the quadratic formula, y = (-x +or- sqrt(x^2 + 4*1*x^2))/2 = x(-1 +or- sqrt(5))/2 The negative solution is spurious since y is a length, so y/x = (sqrt(5)-1)/2 The base of the big isosceles triangle is x+y: x+y = x(1+y/x) = x(sqrt(5)+1)/2 Now, draw a perpendicular from B to AC. It bisects AC, so we have a right triangle with hypotenuse x and leg (x+y)/2. Therefore the cosine of angle BAC is cos(BAC) = (x+y)/(2x) = (sqrt(5)+1)/4 Angle BAC is 36 degrees. The sine of 36 degrees can now be found using the Pythagorean theorem. There is probably a neater way to derive this, but it's what I come up with on the fly. Our Dr. Math Archives can tell you more about pentagons, their construction, and the ratio (x+y)/x, which happens to be the Golden Ratio. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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