Exploring the Value of a Trigonometric Product: sin^2 180/40 sin^2 2180/40 ... sin^2 20180/40

The problem presented here involves evaluating the product of trigonometric functions: sin^2 180/40 sin^2 2180/40 sin^2 3180/40 ... sin^2 20180/40. This can be approached using trigonometric identities and properties of periodic functions to simplify the expression and find its value.

Introduction to the Problem

We start by assuming that all angles are in degrees. The index of the trigonometric function, 180/40, can be simplified to a more manageable form: x 4.5 deg. We need to evaluate the product of the square of sine for angles increasing by 4.5 degrees:

S sin^2 (4.5) sin^2 (9) sin^2 (13.5) ... sin^2 (20180) / 40

Simplification using Trigonometric Identities

The expression can be simplified using trigonometric identities. We can express the square of the sine function in terms of the cosine function:

S (1 - cos 2x)/2 * (1 - cos 4x)/2 * (1 - cos 6x)/2 ... (1 - cos 4020)/2

This can be further simplified as:

S 1/2 * (20 - (cos 2x cos 38x cos 4x cos 36x ... cos 18x cos 22x))

Evaluating the Cosine Products

Next, we need to evaluate the cosine products. We know that cosine is a periodic function with a period of 360 degrees, and it has specific values for multiples of 90 degrees. Let's use this property to simplify the expression.

We start by noting that:

cos 2 cos (20 degrees) 0

and

cos 4 cos (40 degrees) -1

Therefore, the expression simplifies as follows:

S 1/2 * (20 - (0 -1) - 2cos(2)(cos(8) cos(16) ... cos(36)))

We further simplify the expression to:

S 1/2 * (20 1 - 2cos(2)(2cos(2)(...)))

Since all other cosine terms will be 0 due to the periodic property of cosine and the presence of 90-degree intervals, we get:

S 1/2 * (21) 10.5

Thus, the value of the trigonometric product is 10.5.

Conclusion

This solution demonstrates how to simplify and evaluate a trigonometric product using basic trigonometric identities and the periodic property of cosine. The final value of the given product is 10.5, highlighting the power of trigonometric identities in solving complex mathematical problems.

References

1. Wikipedia: List of Trigonometric Identities

2. Math is Fun: Trigonometry