Find the values of the following allied angles :
(v) `sin 330^(@)`

Answer

To find the value of \( \sin 330^\circ \), we can follow these steps:

### Step 1: Identify the angle in relation to the unit circle
The angle \( 330^\circ \) is located in the fourth quadrant of the unit circle.

### Step 2: Rewrite the angle using a reference angle
We can express \( 330^\circ \) in terms of a reference angle. The reference angle for \( 330^\circ \) is:
\[ 360^\circ - 330^\circ = 30^\circ \]
Thus, we can write:
\[ \sin 330^\circ = \sin(360^\circ - 30^\circ) \]

### Step 3: Use the sine subtraction identity
Using the sine subtraction identity, we know:
\[ \sin(360^\circ - \theta) = -\sin(\theta) \]
Applying this to our angle:
\[ \sin(360^\circ - 30^\circ) = -\sin(30^\circ) \]

### Step 4: Find the sine of the reference angle
Now, we need to find \( \sin(30^\circ) \). From trigonometric values, we know:
\[ \sin(30^\circ) = \frac{1}{2} \]

### Step 5: Substitute back to find \( \sin 330^\circ \)
Substituting back into our equation:
\[ \sin 330^\circ = -\sin(30^\circ) = -\frac{1}{2} \]

### Final Answer
Thus, the value of \( \sin 330^\circ \) is:
\[ \sin 330^\circ = -\frac{1}{2} \]

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