In triangle ABC, if `A - B = 120^(2) and R = 8r`, where R and r have their usual meaning, then cos C equals

Options

A. `3//4`
B. `2//3`
C. `5//6`
D. `7//8`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We are given that in triangle ABC, \( A - B = 120^\circ \) and \( R = 8r \), where \( R \) is the circumradius and \( r \) is the inradius.

### Step 2: Use the formula for inradius
The formula for the inradius \( r \) of a triangle is given by:
\[ r = \frac{A}{s} \]
where \( A \) is the area of the triangle and \( s \) is the semi-perimeter. However, we will use the relation involving angles:
\[ r = \frac{4R \cdot \sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2}}{s} \]
Given \( R = 8r \), we can express \( r \) in terms of \( R \):
\[ r = \frac{R}{8} \]

### Step 3: Substitute \( r \) in the inradius formula
Substituting \( r \) into the inradius formula, we have:
\[ \frac{R}{8} = \frac{4R \cdot \sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2}}{s} \]
Cancelling \( R \) from both sides (assuming \( R \neq 0 \)):
\[ \frac{1}{8} = \frac{4 \cdot \sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2}}{s} \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ s = 32 \cdot \sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2} \]

### Step 5: Use the identity for \( \sin \frac{A}{2} \) and \( \sin \frac{B}{2} \)
Using the identity:
\[ \sin \frac{A}{2} \cdot \sin \frac{B}{2} = \frac{1}{2} \left( \cos \frac{A-B}{2} - \cos \frac{A+B}{2} \right) \]
We know \( A - B = 120^\circ \), thus:
\[ \cos \frac{A-B}{2} = \cos 60^\circ = \frac{1}{2} \]
And since \( A + B + C = 180^\circ \), we have:
\[ A + B = 180^\circ - C \implies \cos \frac{A+B}{2} = \sin \frac{C}{2} \]

### Step 6: Substitute values into the equation
Substituting these into our equation gives:
\[ \sin \frac{C}{2} = \frac{1}{2} \left( \frac{1}{2} - \sin \frac{C}{2} \right) \]
Let \( x = \sin \frac{C}{2} \):
\[ x = \frac{1}{4} - \frac{x}{2} \]
Rearranging gives:
\[ \frac{3x}{2} = \frac{1}{4} \implies x = \frac{1}{6} \]

### Step 7: Find \( \cos C \)
Using the identity:
\[ \cos C = 1 - 2 \sin^2 \frac{C}{2} \]
Substituting \( \sin \frac{C}{2} = \frac{1}{4} \):
\[ \cos C = 1 - 2 \left( \frac{1}{4} \right)^2 = 1 - 2 \cdot \frac{1}{16} = 1 - \frac{1}{8} = \frac{7}{8} \]

### Final Answer
Thus, the value of \( \cos C \) is:
\[ \cos C = \frac{7}{8} \]