Period of $\sin(x) + \cos(x)$

Question

The period of $\sin(x)$ is $2\pi$ and $\cos(x)$ is $2\pi$.

And the period of $\sin(x)+\cos(x)$ is also $2\pi$.

Why it is so?

Can you clarify what you mean by why? How are you defining $\sin$ and $\cos$? — Mathmo123, Jul 20 '14 at 18:29
Suppose the period of a function $f(x)$ is $h$. Then, by the definition of the period of a function, we have $f(x+h)=f(x)$. How does this work out when we have a sum of functions $f(x),g(x)$? — abiessu, Jul 20 '14 at 18:30

score 9 · Answer 1 · answered Jul 20 '14 at 18:39

9

There is something different about the function $$\sin x+\cos x= \sqrt 2(\frac{\sqrt 2}2\sin x+\frac {\sqrt 2} 2\cos x)=\sqrt 2(\cos \frac {\pi}4\sin x+\sin\frac {\pi}4\cos x)=\sqrt 2\sin (x+\frac {\pi}4)$$- it is not the period of the function, which remains $2\pi$, but the amplitude.

answered Jul 20 '14 at 18:39

Mark Bennet

101,769

4

This is the better way because it proves that the period cannot be smaller than $2\pi $ as well. – Matt B. Jul 20 '14 at 18:42
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Perhaps I should also have mentioned that the $\frac {\pi}4$ is a phase shift. – Mark Bennet Jul 20 '14 at 18:44

score 2 · Answer 2 · answered Jul 20 '14 at 18:35

To find the period of a function:

Given $$\sin(kx)$$ or $$\cos(kx)$$

So, the period will be $$\frac{2\pi}{k}$$

Now for $$\cos x + \sin x$$

Now, see that we must have an integral number of periods between $\sin x$ and $\cos x$

So, for positive integers $m$ and $n$:

$$2\pi m = 2\pi n$$

This can be satisfied if $m = n =1$

So the period is thus $2\pi$

score 1 · Answer 3 · answered Jul 20 '14 at 18:33

1

Let $f(x) = \sin(x) + \cos(x)$. Then

$$f(x + 2\pi) = \sin(x + 2\pi) + \cos(x + 2\pi) = \sin(x) + \cos(x) = f(x)$$

answered Jul 20 '14 at 18:33

naslundx

9,896

score 1 · Answer 4 · answered Jul 20 '14 at 18:33

1

because $$\sin(x+2\pi)+\cos(x+2\pi)=\sin(x)+\cos(x)$$

answered Jul 20 '14 at 18:33

Adi Dani

17,007

2

I see this as a proof that $2\pi$ is a period. The OP speaks of the period. It must be shown also that values between $0$ and $2\pi$ are not periods. – drhab Jul 20 '14 at 18:41
By the same logic, $\sin(x)\cos(x)$ would have a period of $2\pi$ as well. But it doesn't; it has a period of $\pi$. (Proof: Remember that $\sin(x)\cos(x)=\frac12\sin(2x)$) – Akiva Weinberger Mar 27 '15 at 18:02

Period of $\sin(x) + \cos(x)$

4 Answers4