Express the following as functions of angles less than `45^(@)` :
`sin(-1785^(@))`

Answer

To express \( \sin(-1785^\circ) \) as a function of angles less than \( 45^\circ \), we can follow these steps:

### Step 1: Use the periodic property of sine
The sine function is periodic with a period of \( 360^\circ \). This means:
\[ \sin(\theta) = \sin(\theta + 360^\circ n) \]
for any integer \( n \).

### Step 2: Reduce the angle
To find an equivalent angle for \( -1785^\circ \), we can add \( 360^\circ \) repeatedly until we get an angle within the range of \( 0^\circ \) to \( 360^\circ \):
\[ -1785 + 360 \times n \]
We can calculate how many times we need to add \( 360^\circ \):
\[ n = \left\lceil \frac{1785}{360} \right\rceil = 5 \]
Calculating:
\[ -1785 + 360 \times 5 = -1785 + 1800 = 15^\circ \]
Thus, we have:
\[ \sin(-1785^\circ) = \sin(15^\circ) \]

### Step 3: Use the property of sine for negative angles
We know that:
\[ \sin(-\theta) = -\sin(\theta) \]
So, we can express:
\[ \sin(-1785^\circ) = \sin(15^\circ) = -\sin(15^\circ) \]

### Step 4: Final expression
Since \( 15^\circ < 45^\circ \), we can conclude:
\[ \sin(-1785^\circ) = \sin(15^\circ) \]

### Final Answer
Thus, the expression for \( \sin(-1785^\circ) \) as a function of angles less than \( 45^\circ \) is:
\[ \sin(-1785^\circ) = \sin(15^\circ) \]

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