Trigonometry and the Golden Ratio

topic posted Sat, April 1, 2006 - 11:00 PM by  Unsubscribed
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Looking to sharpen your knowledge of the golden ratio?

If you have some experience with compass/ruler constructions try this-

Determine the trigonometric values of angles such as 18, 36, 54, and 72 degreees.
Express them in terms of the Golden Ratio and also express them in their simplest radical form.

Hint: You will find these angles in a regular pentagon. You will need to rigorously construct a pentagon. If you havn't done so - try some simpler constructions first.
Arm yourself with a true understanding of the Pythagorean Theorem. Know some basic algebra so you can simplify expressions.
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  • Re: Trigonometry and the Golden Ratio

    Fri, January 11, 2008 - 6:57 AM
    I've found out that the ratio of the length of a line linking two non-adjacent vertices of a regular pentagon to the length of its side is the golden ratio.

    Hence the cosine of 36 degrees is half the golden ratio (about 0.809).

    Karl
    • Unsu...
       

      Re: Trigonometry and the Golden Ratio

      Sun, January 13, 2008 - 9:25 PM
      In radians 36 degrees is pi/5.
      If you look at a table of so called "special angles" in a textbook
      you should notice that the exact values of angles pi/2, pi/3, pi/4,
      pi/6 (90, 60, 45, and 30 degrees) are listed but not pi/5. So now you have it.

      cos(pi/5)=phi/2=(1+sqrt.5)/4
      which is also equal to sin(3*pi/10), the complimentary angle. (3*pi/10 = 54 degrees)

      And with Pythagorean theorem you can find the sin(pi/5) which
      also equals cos(3*pi/10)

      Trig textbooks also illustrate "special triangles" ; right triangles with the angles listed in that table.
      The 45-45-90 triangle, and the 30-60-90 triangle.
      So now we have another useful special triangle-
      36-54-90.

      But calculating the sin and cos of the angles is a bit more difficult for beginner trig students.
      It requires understanding pentagons and the golden ratio.

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