“Trigonometry Values Pi/7”的概念、定义、习题解题方法及翻译-数学辞典

单词

Trigonometry Values Pi/7

释义

Trigonometry Values Pi/7

Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite rootextractions on real rational numbers because 7 is not a Fermat Prime. This also means that the Heptagonis not a Constructible Polygon.

However, exact expressions involving roots of complex numbers can still be derived using the trigonometric identity

(1)

The case

gives







	(2)

Rewrite this using the identity


	(3)

Now, let

and

, then

(4)

which is a Cubic Equation in

. The Roots are numerically found to be

. But

, so these Roots correspond to

. By Newton'sRelation

(5)

we have

(6)

(7)

Similarly,

(8)

The constants of the Cubic Equation are given by

			(9)


			(10)

The Discriminant is then


			(11)

so there are three distinct Real Roots. Finding the first one,

(12)

Writing

(13)

plugging in from above, and anticipating that the solution we have picked corresponds to




	(14)

See also Heptagon

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