Without using tables, give the value of each of the following :
`sin120^(@)`

Answer

To find the value of \( \sin 120^\circ \) without using tables, we can follow these steps:

### Step 1: Express \( \sin 120^\circ \) in terms of a known angle
We can express \( 120^\circ \) as \( 90^\circ + 30^\circ \):
\[ \sin 120^\circ = \sin(90^\circ + 30^\circ) \]

### Step 2: Use the sine addition formula
Using the sine addition formula, we know that:
\[ \sin(90^\circ + \theta) = \cos(\theta) \]
Thus, we have:
\[ \sin(90^\circ + 30^\circ) = \cos(30^\circ) \]
So, we can rewrite our expression:
\[ \sin 120^\circ = \cos 30^\circ \]

### Step 3: Find the value of \( \cos 30^\circ \)
To find \( \cos 30^\circ \), we can use the properties of an equilateral triangle. Consider an equilateral triangle with each side of length \( a \). The angles in this triangle are all \( 60^\circ \).

### Step 4: Draw the median
When we draw a median from one vertex to the opposite side, it will bisect the angle and the side. This median will also be perpendicular to the side it meets. Therefore, we can form a right triangle with:
- One angle \( 30^\circ \) (half of \( 60^\circ \))
- The hypotenuse \( a \)
- The base (half of the side) \( \frac{a}{2} \)

### Step 5: Apply the Pythagorean theorem
Using the Pythagorean theorem in this triangle:
\[ \text{Let the height from the vertex to the base be } h. \]
Then:
\[ h^2 + \left(\frac{a}{2}\right)^2 = a^2 \]
This simplifies to:
\[ h^2 + \frac{a^2}{4} = a^2 \]
\[ h^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4} \]
Taking the square root gives:
\[ h = \frac{\sqrt{3}a}{2} \]

### Step 6: Calculate \( \cos 30^\circ \)
Now, in the triangle, \( \cos 30^\circ \) is given by:
\[ \cos 30^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\frac{a}{2}}{a} = \frac{1}{2} \]
However, we need to consider the height we calculated:
\[ \cos 30^\circ = \frac{\text{base}}{\text{hypotenuse}} = \frac{\frac{a}{2}}{a} = \frac{\sqrt{3}}{2} \]

### Conclusion
Thus, we find:
\[ \sin 120^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2} \]

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