Q:

# How is cos 4pi evaluated?

A:

There are two ways to evaluate cos 4? that will both give the answer of 1. The best ways to evaluate involve the periodicity of the cosine function and the trigonometric addition formula for cosine.

## Keep Learning

First, the cosine function has a period of 2?, meaning that cos x = cos (x + 2?). Using this formula, we know that cos 0 = cos 2? = cos 4?. Recalling that cos 0 = 1, we know that cos 4? = cos 0 = 1.

In cases of cos 2n? or sin 2n?, where n is a whole number, this is a simple process. However, this process does not always involve simple terms like this. This is when the trigonometric addition formula for cosine is helpful.

First, suppose that cos z needs to be evaluated, and z = x + y, where x, y and z are real numbers.

Then, cos(x + y) = (cos x)(cos y) -(sin x)(sin y)

Using 2? as both x and y to make z = 4?,

cos 4? = cos (2? + 2?) = (cos 2?)(cos 2?) - (sin 2?)(sin 2?)

Now, recalling that cos 2? = 1 and sin 2? = 0,

cos 4? = (1) (1) - (0) (0) = 1.

While this method involves much more calculation, it works for any angle.

Sources:

## Related Questions

• A: Answers to the June 2010 Algebra 2 and Trigonometry Regents exams are available from the New York State Department of Education's Office of State Assessmen... Full Answer >
Filed Under:
• A: A rectangle has two lines of symmetry. It can be folded horizontally and vertically, from the middle of the sides, but it cannot be folded corner to corner... Full Answer >
Filed Under:
• A: The periods of the trigonometric functions sine and cosine are both 2 times pi. The functions tangent and cotangent both have a period of pi. The general f... Full Answer >
Filed Under:
• A: One of the most common applications of trigonometry in everyday life is the use of triangulation to determine the height of buildings, mountains, trees and... Full Answer >
Filed Under:
PEOPLE SEARCH FOR