Question

In \( \triangle ABC \), if \( a = 18 \), \( b = 24 \), and \( c = 30 \), then find the value of \[ \sin \left(\frac{A}{2} \right). \]

Hint:

To find the sine of half an angle in triangle problems, use the formula: \[ \sin \left(\frac{A}{2}\right) = \sqrt{\frac{(s - b)(s - c)}{b c}}. \]

Answer 1

Step 1: Calculate the Semi-Perimeter
The semi-perimeter \( s \) of the triangle is computed as: \[ s = \frac{a + b + c}{2} = \frac{18 + 24 + 30}{2} = 36. \] Step 2: Apply the Half-Angle Formula for Sine
The half-angle formula for sine is: \[ \sin \left(\frac{A}{2}\right) = \sqrt{\frac{(s - b)(s - c)}{b c}}. \] Step 3: Substituting the Given Values
Substitute the known values into the formula: \[ \sin \left(\frac{A}{2}\right) = \sqrt{\frac{(36 - 24)(36 - 30)}{24 \times 30}}. \] \[ = \sqrt{\frac{12 \times 6}{720}} = \sqrt{\frac{72}{720}} = \sqrt{\frac{1}{10}}. \] \[ = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10}. \]