If a=16, b=24 nad c=20, then the value of `cos(B/2)` is

Options

A. `3/4`
B. `1/4`
C. `1/2`
D. `1/3`

Answer

To find the value of \( \cos \left( \frac{B}{2} \right) \) given \( a = 16 \), \( b = 24 \), and \( c = 20 \), we can use the half-angle formula for cosine:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{s(s-b)}{a \cdot c}} \]

where \( s \) is the semi-perimeter of the triangle defined by sides \( a \), \( b \), and \( c \):

\[ s = \frac{a + b + c}{2} \]

### Step 1: Calculate the semi-perimeter \( s \)

Substituting the values of \( a \), \( b \), and \( c \):

\[ s = \frac{16 + 24 + 20}{2} = \frac{60}{2} = 30 \]

### Step 2: Substitute \( s \) into the half-angle formula

Now we substitute \( s \) into the formula for \( \cos \left( \frac{B}{2} \right) \):

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{30(30 - 24)}{16 \cdot 20}} \]

### Step 3: Simplify the expression

Calculate \( 30 - 24 \):

\[ 30 - 24 = 6 \]

Now substitute this back into the equation:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{30 \cdot 6}{16 \cdot 20}} \]

### Step 4: Calculate the numerator and denominator

Calculating the numerator:

\[ 30 \cdot 6 = 180 \]

Calculating the denominator:

\[ 16 \cdot 20 = 320 \]

Now we have:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{180}{320}} \]

### Step 5: Simplify the fraction

We can simplify \( \frac{180}{320} \):

\[ \frac{180}{320} = \frac{9}{16} \]

### Step 6: Take the square root

Now we take the square root:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### Final Answer

Thus, the value of \( \cos \left( \frac{B}{2} \right) \) is \( \frac{3}{4} \).

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