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Helios
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If tan x = cos x, then what is x ? The answer includes the golden ratio !
Helios said:If tan x = cos x, then what is x ? The answer includes the golden ratio !
adriank said:You forgot the other solution: [tex]\sin^{-1}\bigl[-\tfrac12(1+\sqrt5)\bigr][/tex]. (Not a real solution, though.)
JJacquelin said:k = any negative, null or positive integer.
Char. Limit said:I just realized that [itex]sin^{-1}(\phi)[/itex] isn't real either... wow, so there are actually no real solutions.
Mentallic said:So cos(x) doesn't cross tan(x)?
Char. Limit said:Mistake number two...
The real solution is in fact:
[tex]sin^{-1}\left(\frac{-1}{\phi}\right)[/tex]
The Golden Ratio is a mathematical concept that is often represented by the Greek letter phi (φ). It is approximately equal to 1.618 and is found by dividing a line into two parts, with the longer part divided by the smaller part being equal to the sum of the two original parts.
The Golden Ratio can be used in solving for x by setting up an equation where x is equal to the ratio of two terms in a sequence, such as in the Fibonacci sequence. This can help uncover patterns and relationships between numbers, leading to a solution for x.
Yes, the Golden Ratio can be found in many natural phenomena, such as the arrangement of leaves on a stem, the shape of a seashell, and the proportions of the human body. It is believed that this ratio is aesthetically pleasing and is often used in art and architecture.
The Golden Ratio is closely related to the Fibonacci sequence, where each number is the sum of the two previous numbers (e.g. 1, 1, 2, 3, 5, 8, 13, etc.). As the sequence gets larger, the ratio between consecutive numbers approaches the Golden Ratio.
Yes, the Golden Ratio has been used in various fields such as art, architecture, music, and design. It is believed to create a sense of balance and harmony, and many famous works of art and architecture, such as the Parthenon and the Mona Lisa, are said to follow the proportions of the Golden Ratio.