Evaluate `sin 120^(@)`

Options

A. ` (sqrt(3))/(4)`
B. ` 1/sqrt(2)`
C. ` 1/2`
D. ` (sqrt(3))/(2)`

Explanation

To evaluate \( \sin 120^\circ \), we can follow these steps:

### Step 1: Rewrite \( 120^\circ \)
We can express \( 120^\circ \) as:
\[ 120^\circ = 90^\circ + 30^\circ \]

### Step 2: Use the sine addition formula
Using the sine addition formula, we know that:
\[ \sin(90^\circ + \theta) = \cos(\theta) \]
where \( \theta = 30^\circ \).

### Step 3: Apply the formula
Applying this to our angle:
\[ \sin(120^\circ) = \sin(90^\circ + 30^\circ) = \cos(30^\circ) \]

### Step 4: Find \( \cos(30^\circ) \)
Now we need to find the value of \( \cos(30^\circ) \):
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

### Step 5: Conclusion
Thus, we can conclude that:
\[ \sin(120^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

### Final Answer
\[ \sin(120^\circ) = \frac{\sqrt{3}}{2} \]

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