In `triangleABC`, if `a=18, b=24, c=30`, then `cos((A)/(2))=`

Options

A. `(4)/(sqrt(10))`
B. `(3)/(sqrt(10))`
C. `(2)/(sqrt(10))`
D. `(1)/(sqrt(10))`

Answer

To find \( \cos\left(\frac{A}{2}\right) \) in triangle \( ABC \) where \( a = 18 \), \( b = 24 \), and \( c = 30 \), we can follow these steps:

### Step 1: Calculate the semi-perimeter \( s \)
The semi-perimeter \( s \) of triangle \( ABC \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ s = \frac{18 + 24 + 30}{2} = \frac{72}{2} = 36 \]

### Step 2: Use the formula for \( \cos\left(\frac{A}{2}\right) \)
The formula for \( \cos\left(\frac{A}{2}\right) \) is:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{s(s - a)}{bc}} \]

### Step 3: Substitute the values into the formula
Now we substitute \( s = 36 \), \( a = 18 \), \( b = 24 \), and \( c = 30 \):
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{36(36 - 18)}{24 \cdot 30}} \]

### Step 4: Simplify the expression
First, calculate \( s - a \):
\[ s - a = 36 - 18 = 18 \]
Now substitute this back into the equation:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{36 \cdot 18}{24 \cdot 30}} \]

### Step 5: Calculate the denominator
Calculate \( 24 \cdot 30 \):
\[ 24 \cdot 30 = 720 \]

### Step 6: Calculate the numerator
Calculate \( 36 \cdot 18 \):
\[ 36 \cdot 18 = 648 \]

### Step 7: Substitute back into the equation
Now we have:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{648}{720}} \]

### Step 8: Simplify the fraction
To simplify \( \frac{648}{720} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 72:
\[ \frac{648 \div 72}{720 \div 72} = \frac{9}{10} \]

### Step 9: Take the square root
Now we take the square root:
\[ \cos\left(\frac{A}{2}\right) = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}} \]

### Final Answer
Thus, the value of \( \cos\left(\frac{A}{2}\right) \) is:
\[ \cos\left(\frac{A}{2}\right) = \frac{3}{\sqrt{10}} \]