Where in mathematics is the golden ratio used? And find one worked example or proof if possible?

Question

i need this answer for a report and it would be good if i can get this answer before the break
thxx:)

Answer 1

Well one example is in trigonometry, it allows you to find the exact value of cos 72°. Here's how:

Let ABC be a 72°-72°-36° triangle, with ∠C=36°. Draw the bisector of ∠B, and let it intersect AC at point D. By the definition of angle bisector, this makes ∠DBC = 36°. Since ∠BCD = 36°, we have that BCD is isoceles with CD≡DB. Now ∠DBA = 36°, and ∠DAB = 72°, so the third angle in ADB, which is ∠BDA, must equal 180°-36°-72°, which is 72°, so ADB is isoceles with BD≡BA. Now, ADB and ABC are both 72°-72°-36° triangles, so ADB ~ ABC, and it follows that AC/AB = AB/AD. But we previously established that AB≡BD≡DC, so we have AC/AB = AC/DC = DC/AD. Clearly, AC = AD+DC, so we have AC/AB = (AD+DC)/DC = DC/AD. Replacing AD with a and DC with b, we have that this ratio is the rato given by (a+b)/b = b/a, which is precisely the golden ratio.

To compute the golden ratio, we set φ = b/a. Then we have:

a/b + b/b = b/a
1/φ + 1 = φ
1+φ = φ²
φ² - φ - 1 = 0

By the quadratic formula:

φ = (1±√5)/2

But we know that φ is a ratio of lengths, which are both positive, so we may safely discard the negatvie solution.

Having shown that AC/AB = φ, we then use this known ratio of sides to compute the cosine. We draw an altitude from C onto AB at P. Since this is an isoceles triangle, this bisects AB, so AP = AB/2. Further, this forms right triangle APC, with ∠A=72°, so we have that cos 72° = AP/AC = 1/2 AB/AC = 1/(2φ) = 1/(1+√5) = (√5-1)/4. This, combined with the better-known identities for angles of 60°, 45°, and 30°, and the half-angle formulae, allows you to compute exact values of sin and cos for any multiple of 3°.

Answer 2

I believe the ancient Greeks used the golden ratio in the design of many of their buildings, such as the Parthenon. Artists and other creative types have also used this ratio in their work through the centuries.

I don't know about a proof, but there is a method used to find the golden ratio. First, start with any linear piece which you designate as having length equal to x units. It doesn't matter what the units are. Then divide the piece into two smaller pieces which you designate 1 unit and x-1 units. The object is to divide the piece so that the ratio of the entire length, x, to the piece of length 1 is equal to the ratio of the piece of length 1 to the piece of length x-1. Mathematically we write it this way:

x/1 = 1/x-1

We multiply this out to come up with this equation:

x^2-x = 1 --> x^2-x -1 = 0.

Then we solve for x:

x = [1 +or- sq rt (5)]/2.

It is the sum of the two terms in the numerator we are interested in.

x = (1 + 2.236067977...)/2 = 3.236.../2 = 1.618033989...

Actually the above is an approximation because the square root of 5 is an irrational number.

The ancient Greeks, it is said, tried to construct many structures so that the ratio of the width to the height of its facade approximated this ratio. They did this because it was said to be aesthetically pleasing to the eye. Undoubtedly, knowing the Greeks, it also probably had some significance to one of their many gods.

The golden ratio has significance in mathematics because the Fibonacci sequence approaches this ratio as the number of terms approaches infinity. In this sequence, each term is the sum of the two preceding terms and it is actually the ratio of the last two terms which we compute.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

Notice that after only 13 terms the ratio (1.6180555...) is already very close to the golden ratio.

In nature, plants like daisies, sunflowers, and pinecones are said to exhibit the golden ratio in the arrangement of their parts. The nautilus shell also grows in a spiral equal to the golden ratio. The human body is also said to be proportioned according to the golden ratio.

Answer 3

The golden ratio is the coolest thing ever! If you compare adjacent terms of the Fibionacci sequence, the ratio approaches the golden ratio as you go further on in the sequence. That is to say, the Fibionacci sequence becomes almost geometric. Yet the Fibionacci sequence is generated by addition!! In nature, you see the Fibionacci sequence in anything that grows at a rate in discrete units based on the area that's already there, such as the arrangement of spikes on a pinecone, florets on the head of a composite flower, or successive radii of certain seashells. This seems to be related to how a sequence that is generated by addition yet becomes "almost" geometric. The Golden Ratio thus seems to be one of the keys to creation, which is why it is sometimes called the Divine Proportion.

The golden ratio is the solution to Aristotle's description of perfect proportion, that the width: length = length : whole. Translate this into a quadratic equation and solve and you get (1 +/- sqrt 5)/2

Here are some suggestions for your report. Write out successive values of the Fibonacci sequence, and compare their ratios. Write out the quadratic equation whose solution is the golden ratio, and show why it fit's Aristotle's description. Try it yourself!

Answer 4

The golden ratio is the best number ever!! First of all, you could say that the sine of 666 is -1/2 of the golden ratio (the sign of the devil is the opposite of good but only half as strong). Also, it is used in the fibonacci sequence. Just look it up!!!!

Answer 5

Check into regular pentagons. For example, it is the ratio of the length of a diagonal to the length of a side.