Evaluating Trigonometric Sums Using Lagrange's Identity and Cosine Squared

When evaluating trigonometric sums, one powerful tool is Lagrange's identity. This identity allows us to derive elegant formulas for sums involving cosine and sine functions. In this article, we will explore how to use Lagrange's identity to evaluate sums of cosine and sine terms in the context of degrees. We'll engage with various mathematical concepts and demonstrate step-by-step derivations to help you grasp the underlying principles.

Understanding the Basics

First, let's recall that the fundamental trigonometric identity involves the sum of cosines and sines. The key relationship in this context is:

Lagrange's Identity

Lagrange's identity for a sum of cosines is given by:

[ sum_{i0}^{n} cos itheta frac{sinleft(frac{(2n-1)theta}{2}right)}{2sinleft(frac{theta}{2}right)} ]

Using Lagrange's identity, the sum of squares of cosines can be expressed in a simplified form, leading to the formula for a specific case involving degrees.

Applying Lagrange's Identity

Given the problem of finding the sum of [ sum_{k1}^{44} cos^2 kfrac{pi}{180} sin^2 kfrac{pi}{180} ], we start by using the identity for the sum of cosines squared:

[ S_n sum_{k1}^{n} cos^2 k^circ frac{1}{2} sum_{k1}^{n} cos 2k^circ - frac{n}{2} ]

By converting degrees to radians, we have:

[ sum_{k1}^{n} cos 2k^circ sum_{k1}^{n} cos left(frac{pi}{90}kright) frac{sinleft(frac{2n-1}{90}piright)}{2sinleft(frac{pi}{180}right)} - frac{1}{2} ]

Substituting this back into the formula for (S_n), we get:

[ S_n frac{n}{2} cdot frac{sinleft(frac{2n-1}{180}piright)}{2sinleft(frac{pi}{180}right)} - frac{n}{4} ]

Note that the specific value of the sum for (n44) will be derived from this general formula.

Example Problem

Problem: Evaluate the Sum

Evaluate the sum ( sum_{k1}^{44} cos^2 kfrac{pi}{180} sin^2 kfrac{pi}{180} ).

Solution:

First, we use the general formula derived from Lagrange's identity:

[ sum_{k1}^{n} cos^2 k^circ frac{1}{2} sum_{k1}^{n} cos 2k^circ - frac{n}{2} ]

Converting to radians, we have:

[ sum_{k1}^{n} cos 2k^circ sum_{k1}^{n} cos left(frac{pi}{90}kright) frac{sinleft(frac{2n-1}{90}piright)}{2sinleft(frac{pi}{180}right)} - frac{1}{2} ]

Thus, the sum becomes:

[ S_{44} frac{44}{2} cdot frac{sinleft(frac{87pi}{180}right)}{2sinleft(frac{pi}{180}right)} - frac{44}{4} ]

Simplifying further based on the properties of sine and the symmetry of the cosine function, the exact numerical value can be calculated.

Conclusion

In this article, we explored the process of evaluating sums of trigonometric functions using Lagrange's identity. We demonstrated how to convert degrees to radians and applied the identity to simplify the calculation. With a clear understanding of the process, you can now tackle similar problems with confidence. Using Lagrange's identity effectively is a valuable tool in the field of trigonometry and can greatly simplify complex sum evaluations.