Find the value of `sin30^(@)` geometrically.

Answer

To find the value of \( \sin 30^\circ \) geometrically, we can follow these steps:

### Step 1: Draw an Equilateral Triangle
Draw an equilateral triangle \( ABC \) where each side is of length \( A \).

### Step 2: Draw the Perpendicular Bisector
Draw the perpendicular bisector \( AD \) from vertex \( A \) to the midpoint \( D \) of side \( BC \). This line \( AD \) is perpendicular to \( BC \).

### Step 3: Identify Angles
Since triangle \( ABC \) is equilateral, each angle is \( 60^\circ \). The line \( AD \) bisects angle \( A \), so angle \( BAD \) is \( 30^\circ \) (i.e., \( 60^\circ / 2 \)).

### Step 4: Analyze Triangle \( ADC \)
In triangle \( ADC \):
- The hypotenuse \( AC \) is equal to \( A \).
- The side opposite angle \( 30^\circ \) (which is \( AD \)) is half of \( BC \), so it is \( \frac{A}{2} \).

### Step 5: Apply the Sine Definition
Using the definition of sine:
\[ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AD}{AC} \]
Substituting the lengths we identified:
\[ \sin(30^\circ) = \frac{\frac{A}{2}}{A} \]

### Step 6: Simplify the Expression
Simplifying the expression gives:
\[ \sin(30^\circ) = \frac{1}{2} \]

### Conclusion
Thus, the value of \( \sin 30^\circ \) is \( \frac{1}{2} \).

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