November 2017 Archives

The dictionary defines the trigonometric functions as ratios between the sides of a right triangle, and in this form they are most frequently taught and applied to physical problems. However, their significance in the natural world transcends this definition - the sine and cosine ratios are central to mathematical descriptions of harmonic motion and oscillations, from the period of a pendulum to the rhythm of a bird's wings.

In the eighteenth century, mathematician Leonhard Euler discovered an elegant formula that showed the sine function to be an infinite-degree polynomial. This realization began with the well-known fact that a polynomial of degree n always possesses n roots, either real or complex. Since the sinusoidal functions have an infinite number of both real and complex roots, it follows that they could be expressed as infinite products. Euler proved that for some constant A, sin(x) could be found using the equation

sinea.pngInterestingly, Euler proved the feasibility of this formula before he found the value of the constant A itself, but in the following months he found a solution. Since the sine of x divided by x itself approaches 1 as x approaches zero, he factored out x from this limit and was left with the true value of A:

valuea.pngThough this expression of sin(x) is seldom taught, it sheds a new perspective on the nature of trigonometric functions, and contributed to important breakthroughs such as the solution of the Basel problem in 1734.

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