If `theta=-1000^(@)`, determine the sign of `theta-cos theta`.

Answer

To determine the sign of \( \theta - \cos \theta \) where \( \theta = -1000^\circ \), we can follow these steps:

### Step 1: Substitute the value of \( \theta \)
We start with the expression:
\[ \theta - \cos \theta \]
Substituting \( \theta = -1000^\circ \):
\[ -1000^\circ - \cos(-1000^\circ) \]

### Step 2: Simplify \( \cos(-1000^\circ) \)
Using the property of cosine that \( \cos(-x) = \cos(x) \), we can rewrite:
\[ -1000^\circ - \cos(1000^\circ) \]

### Step 3: Find the equivalent angle for \( 1000^\circ \)
To find \( \cos(1000^\circ) \), we need to reduce \( 1000^\circ \) to an angle within the standard range of \( [0^\circ, 360^\circ] \). We do this by subtracting \( 360^\circ \) until we are within this range:
\[ 1000^\circ - 2 \times 360^\circ = 1000^\circ - 720^\circ = 280^\circ \]
Thus, \( \cos(1000^\circ) = \cos(280^\circ) \).

### Step 4: Determine the value of \( \cos(280^\circ) \)
The angle \( 280^\circ \) lies in the fourth quadrant where cosine is positive. The reference angle for \( 280^\circ \) is:
\[ 360^\circ - 280^\circ = 80^\circ \]
Thus, \( \cos(280^\circ) = \cos(80^\circ) \) which is positive.

### Step 5: Substitute back into the expression
Now we can substitute back:
\[ -1000^\circ - \cos(1000^\circ) = -1000^\circ - \cos(280^\circ) \]
Since \( \cos(280^\circ) > 0 \), we can denote it as \( c \) where \( c = \cos(280^\circ) > 0 \):
\[ -1000^\circ - c \]

### Step 6: Determine the sign of the expression
Since \( -1000^\circ \) is a large negative number and \( c \) is a positive number, the overall expression:
\[ -1000^\circ - c \]
will still be negative. Therefore, we conclude that:
\[ \theta - \cos \theta < 0 \]

### Final Answer
The sign of \( \theta - \cos \theta \) is negative.

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