We are given the function:

$f(x) = \lfloor \sin x \rfloor + \lfloor \cos x \rfloor$

where $\lfloor . \rfloor$ represents the greatest integer function (or floor function). To determine the range of $f(x)$, we need to analyze the behavior of $\lfloor \sin x \rfloor$ and $\lfloor \cos x \rfloor$, since the sine and cosine functions take values between $-1$ and $1$.

Step 1: Analyze the possible values of $\lfloor \sin x \rfloor$

• The sine function $\sin x$ takes values in the interval $[-1, 1]$ for all $x$.
• Therefore, $\lfloor \sin x \rfloor$ will take values from the set $\{-1, 0\}$, because:
  • If $\sin x \in [0, 1)$, then $\lfloor \sin x \rfloor = 0$.
  • If $\sin x \in [-1, 0)$, then $\lfloor \sin x \rfloor = -1$.
  • At $\sin x = 1$, $\lfloor \sin x \rfloor = 0$.

Step 2: Analyze the possible values of $\lfloor \cos x \rfloor$

• The cosine function $\cos x$ also takes values in the interval $[-1, 1]$ for all $x$.
• Therefore, $\lfloor \cos x \rfloor$ will also take values from the set $\{-1, 0\}$, because:
  • If $\cos x \in [0, 1)$, then $\lfloor \cos x \rfloor = 0$.
  • If $\cos x \in [-1, 0)$, then $\lfloor \cos x \rfloor = -1$.
  • At $\cos x = 1$, $\lfloor \cos x \rfloor = 0$.

Step 3: Possible values of $f(x)$

Now we combine the two possible values from $\lfloor \sin x \rfloor$ and $\lfloor \cos x \rfloor$. The sum $f(x) = \lfloor \sin x \rfloor + \lfloor \cos x \rfloor$ can take the following values:

• If $\lfloor \sin x \rfloor = 0$ and $\lfloor \cos x \rfloor = 0$, then $f(x) = 0 + 0 = 0$.
• If $\lfloor \sin x \rfloor = -1$ and $\lfloor \cos x \rfloor = 0$, then $f(x) = -1 + 0 = -1$.
• If $\lfloor \sin x \rfloor = 0$ and $\lfloor \cos x \rfloor = -1$, then $f(x) = 0 + (-1) = -1$.
• If $\lfloor \sin x \rfloor = -1$ and $\lfloor \cos x \rfloor = -1$, then $f(x) = -1 + (-1) = -2$.

Thus, the possible values for $f(x)$ are $\{-2, -1, 0\}$.

Final Answer:

The range of $f(x)$ is $\{-2, -1, 0\}$.

Do you need any more details or have further questions?

Here are some related questions to expand the topic:

1. What is the range of $f(x) = \lfloor \sin x \rfloor + \lfloor \tan x \rfloor$?
2. How does the range of $f(x) = \lfloor \sin x \rfloor \times \lfloor \cos x \rfloor$ change?
3. What is the behavior of $f(x) = \lceil \sin x \rceil + \lfloor \cos x \rfloor$, where $\lceil . \rceil$ is the ceiling function?
4. How does the range change if $f(x) = [x] + \sin(x)$?
5. What would be the maximum value of $\lfloor \sin x \rfloor + \lfloor \cos x \rfloor$ on the interval $[0, 2\pi]$?

Tip: The greatest integer function $\lfloor x \rfloor$ returns the largest integer less than or equal to $x$, which simplifies many trigonometric problems!