What is the Value of Cos(nπ/2)?

The value of the cosine function, denoted as (cos(nfrac{pi}{2})), can be understood through the behavior of the cosine at specific angles. The cosine function is periodic with a period of (2pi). This means that the function repeats its values every (2pi) radians. If (n 1, 2, 3, dots), then (cos(nfrac{pi}{2})) can be evaluated at angles that are multiples of (pi/2) radians, which correspond to 90 degrees or 270 degrees.

At these angles, the value of the cosine function is zero. Specifically, when (nfrac{pi}{2}) is a multiple of (pi/2), the cosine value is zero. For example, (cos(frac{pi}{2}) 0), (cos(pi) -1, cos(frac{3pi}{2}) 0,) and (cos(2pi) 1).

Periodicity of the Cosine Function

More generally, the cosine function has a periodic property where (cos(npi 2pi) cos(npi)). This periodicity can be utilized to simplify the evaluation of (cos(nfrac{pi}{2})). For integers (n), (cos(npi)) can be evaluated as follows:

Using Euler's Formula

For a more mathematical insight, we can use Euler's formula, which states that (e^{ipi n} (e^{ipi})^n (-1)^n). This relationship connects complex exponentials with trigonometric functions. Another identity that can be derived from Euler's formula is the cosine function in terms of imaginary exponentials:

(cos(nx) frac{e^{inx} e^{-inx}}{2})

Setting (x pi), we get:

(cos(npi) frac{e^{inpi} e^{-inpi}}{2} frac{(-1)^n (-1)^{-n}}{2} -1^n)

This leads to a more compact expression for the cosine function, demonstrating that for any integer (n), (cos(npi) -1^n).

Conclusion

In summary, the value of the cosine function at integer multiples of (pi) follows specific patterns based on the periodicity of the cosine function and Euler's formula. Understanding these patterns is crucial for applications in various fields, including physics, engineering, and computer science.

For further reading and detailed proofs, consider exploring resources on trigonometry and complex analysis.