Understanding the Value of cos(nπ - 1)

Understanding the behavior of trigonometric functions, particularly the cosine function, is crucial in various fields including mathematics, physics, and engineering. This article delves into the unique property of the cosine function when it involves integer multiples of π, specifically the value of cos(nπ - 1).

Introduction to Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions used in various scientific and engineering applications. The cosine function, in particular, has unique properties when its argument is an integer multiple of π. This article will explore the value of cos(nπ - 1) for integer values of n.

Properties of the Cosine Function

The cosine function, denoted as cos(θ), possesses the property that for any integer k, (cos(kpi) -1^k). This property can be extended to understanding the value of cos(nπ - 1) for integer values of n).

Deriving the Value of cos(nπ - 1)

The key to understanding the value of cos(nπ - 1) lies in recognizing the recursive relationship:

[cos((n - 1)pi) -1^{(n - 1)}]

This relationship can be derived from the cosine function's periodicity and the identity (cos(kpi pi) -cos(kpi)).

Proof by Induction

Let's prove this relationship by induction. For the base case, when n 0, we have:

[cos(-pi) cos(pi) -1 -1^1 -1^{0-1}]

For the inductive step, assume the statement holds for some integer k, i.e., (cos(kpi - pi) -1^{k-1}). We need to show it holds for k 1:

[cos((k 1)pi - pi) -1^{(k 1 - 1)} -1^k]

This confirms that the property holds for any integer n).

Specific Cases

When n is even:

[cos((2m)pi - pi) -1^{(2m - 1)} 1]

When n is odd:

[cos((2m 1)pi - pi) -1^{(2m 1 - 1)} -1]

In any case, the value of cos(nπ - 1) is given by ((-1)^{n-1}) where n is any integer.

Conclusion

The behavior of the cosine function when its argument is an integer multiple of π minus 1 is summarized by the formula (cos(npi - 1) (-1)^{n-1}). This property is a fundamental aspect of trigonometric functions and has broad applications in mathematics and engineering.

Remember the general property of the cosine function: (cos(kpi) (-1)^k).