ABC is an acute-angled triangle incribed in a circle of radius 5 cm and centre O . The sine of angle A is equal to `(3)/(5).` Calculate without using tables :
(i) the length of BC
(ii) sin OBC
(iii) sin BOC
(iv) cos BOC

Answer

To solve the problem step by step, we will calculate the required values based on the given information about triangle ABC inscribed in a circle of radius 5 cm with the sine of angle A being \( \frac{3}{5} \).

### Step 1: Calculate the Length of BC
1. **Given**: Radius \( R = 5 \) cm and \( \sin A = \frac{3}{5} \).
2. **Using the formula for the length of side BC** in a triangle inscribed in a circle:
\[ BC = 2R \sin A \]
Substituting the values:
\[ BC = 2 \times 5 \times \frac{3}{5} = 6 \text{ cm} \]

### Step 2: Calculate \( \sin OBC \)
1. **Using the relationship between angles** in triangle OBC:
\[ \angle BOC = 2A \]
Since \( \angle OBC = \angle OCB \), we can denote \( \angle OBC = x \).
Thus, we have:
\[ \angle BOC + 2x = 180^\circ \implies 2A + 2x = 180^\circ \implies x = 90^\circ - A \]
2. **Using the sine of the angle**:
\[ \sin OBC = \sin(90^\circ - A) = \cos A \]
3. **Finding \( \cos A \)** using the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \implies \left(\frac{3}{5}\right)^2 + \cos^2 A = 1 \implies \frac{9}{25} + \cos^2 A = 1 \]
\[ \cos^2 A = 1 - \frac{9}{25} = \frac{16}{25} \implies \cos A = \frac{4}{5} \]
4. Thus,
\[ \sin OBC = \frac{4}{5} \]

### Step 3: Calculate \( \sin BOC \)
1. **Using the double angle formula**:
\[ \sin BOC = \sin(2A) = 2 \sin A \cos A \]
2. **Substituting the known values**:
\[ \sin BOC = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25} \]

### Step 4: Calculate \( \cos BOC \)
1. **Using the Pythagorean identity**:
\[ \cos^2 BOC + \sin^2 BOC = 1 \implies \cos^2 BOC + \left(\frac{24}{25}\right)^2 = 1 \]
\[ \cos^2 BOC + \frac{576}{625} = 1 \implies \cos^2 BOC = 1 - \frac{576}{625} = \frac{49}{625} \]
\[ \cos BOC = \frac{7}{25} \]

### Summary of Results
1. Length of \( BC = 6 \) cm
2. \( \sin OBC = \frac{4}{5} \)
3. \( \sin BOC = \frac{24}{25} \)
4. \( \cos BOC = \frac{7}{25} \)