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anemone
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Prove that $\cos \dfrac{\pi}{100}$ is irrational.
Rido12 said:Here is a really cheap proof, probably not what you're looking for, but something quick before I resume my homework :p
To prove that it is irrational, I will show that $\cos \dfrac{\pi}{100}$ can be expressed as a sum of two addends, one of which is irrational. It can be proven that if an addend is irrational, then the sum is also irrational. I will use the identity $\cos\left({A-B}\right)=2\cos\left({A}\right)\cos\left({B}\right)+(-\cos\left({A+B}\right))$.
Solving the two equations $A-B=\frac{\pi}{100}$ and $A+B=\frac{\pi}{4}$, then:$$\cos\left({\frac{\pi}{100}}\right)=\cos\left({\frac{13\pi}{100}-\frac{3\pi}{25}}\right)=2\cos\left({\frac{13\pi}{100}}\right)\cos\left({\frac{3\pi}{25}}\right)+(-\cos\left({\frac{\pi}{4}}\right))$$
Since $\cos\left({\frac{\pi}{4}}\right)$ is irrational (also can be proven), $\cos \dfrac{\pi}{100}$ is too.
I cannot agree on this.Rido12 said:Here is a really cheap proof, probably not what you're looking for, but something quick before I resume my homework :p
To prove that it is irrational, I will show that $\cos \dfrac{\pi}{100}$ can be expressed as a sum of two addends, one of which is irrational. It can be proven that if an addend is irrational, then the sum is also irrational. I will use the identity $\cos\left({A-B}\right)=2\cos\left({A}\right)\cos\left({B}\right)+(-\cos\left({A+B}\right))$.
Solving the two equations $A-B=\frac{\pi}{100}$ and $A+B=\frac{\pi}{4}$, then:$$\cos\left({\frac{\pi}{100}}\right)=\cos\left({\frac{13\pi}{100}-\frac{3\pi}{25}}\right)=2\cos\left({\frac{13\pi}{100}}\right)\cos\left({\frac{3\pi}{25}}\right)+(-\cos\left({\frac{\pi}{4}}\right))$$
Since $\cos\left({\frac{\pi}{4}}\right)$ is irrational (also can be proven), $\cos \dfrac{\pi}{100}$ is too.
An irrational number is a real number that cannot be expressed as a ratio of two integers. This means that it cannot be written as a simple fraction and its decimal representation never ends or repeats in a pattern.
The value of cos (π/100) is approximately 0.99995000.
To prove that a number is irrational, we must show that it cannot be expressed as a ratio of two integers. This can be done through various methods such as proof by contradiction or using the definition of irrational numbers.
Proving that cos (π/100) is irrational is significant because it adds to our understanding of the nature of irrational numbers and their relationship to trigonometric functions. It also has implications in other areas of mathematics, such as number theory and geometry.
Some key steps in proving that cos (π/100) is irrational include showing that the angle π/100 is a rational multiple of π, using the definition of cos to express it as a ratio of sides in a triangle, and then using proof by contradiction to show that this ratio cannot be expressed as a ratio of two integers.