Find the period of the function

[utkarshakash]

Homework Statement

Find period of the function
f(x)=cos(cosx)+cos(sin x)

Homework Equations

The Attempt at a Solution

the period of cosx=sinx=2∏. But here cosx and sinx are itself arguments to cosine function.

 

[Dick]

Homework Statement

Find period of the function
f(x)=cos(cosx)+cos(sin x)

Homework Equations

The Attempt at a Solution

the period of cosx=sinx=2∏. But here cosx and sinx are itself arguments to cosine function.

Ok, then f(x)=f(x+2pi) is definitely true. You just have to figure out is there are any shorter periods. Sketching a graph will help.

 

[utkarshakash]

Ok, then f(x)=f(x+2pi) is definitely true. You just have to figure out is there are any shorter periods. Sketching a graph will help.

OK so how do you graph these kinds of functions manually?

 

[ehild]

Hint: Plot the function for multiples of pi/4.

ehild

 

[utkarshakash]

Hint: Plot the function for multiples of pi/4.

ehild

Ok I plotted functions cos(cosx) and cos(sinx) separately for multiples of pi/4 and got the respectives periods as pi. But the given function is sum of both functions. So I took the LCM of periods of both functions which comes out to be pi. But the correct answer is pi/2.

 

[ehild]

Plot the whole function.
It stays the same if the terms are interchanged. What are cos(cos(x+pi/2)) and cos(sin(x+pi/2)) equal to? Expand the arguments.

ehild

 

[vela]

You might also find the identity
$$\cos a + \cos b = 2\cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right)$$ useful.