The angles of a triangle are in the ratio 3 : 5 : 10, the ratio of the smallest side to the greatest side is

Options

A. `1 : sin10^(@)`
B. `1 : 2 sin 10^(@)`
C. `1 : cos 10^(@)`
D. `1 : 2 cos10^(@)`

Explanation

To solve the problem of finding the ratio of the smallest side to the greatest side of a triangle whose angles are in the ratio 3:5:10, we can follow these steps:

### Step 1: Define the Angles
Let the angles of the triangle be represented as:
- \( A = 3x \)
- \( B = 5x \)
- \( C = 10x \)

### Step 2: Set Up the Equation
Since the sum of the angles in a triangle is always \( 180^\circ \), we can set up the equation:
\[ 3x + 5x + 10x = 180 \]

### Step 3: Solve for \( x \)
Combine like terms:
\[ 18x = 180 \]
Now, divide both sides by 18:
\[ x = 10 \]

### Step 4: Calculate the Angles
Now substitute \( x \) back into the expressions for the angles:
- \( A = 3x = 3 \times 10 = 30^\circ \)
- \( B = 5x = 5 \times 10 = 50^\circ \)
- \( C = 10x = 10 \times 10 = 100^\circ \)

### Step 5: Identify the Smallest and Greatest Angles
From the calculated angles:
- The smallest angle is \( A = 30^\circ \).
- The greatest angle is \( C = 100^\circ \).

### Step 6: Use the Law of Sines to Find the Ratio of Sides
According to the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Where \( a \), \( b \), and \( c \) are the sides opposite to angles \( A \), \( B \), and \( C \) respectively.

### Step 7: Express the Ratio of the Smallest Side to the Greatest Side
We want to find the ratio \( \frac{a}{c} \):
\[ \frac{a}{c} = \frac{\sin A}{\sin C} \]

### Step 8: Substitute the Values of the Angles
Substituting the values:
\[ \frac{a}{c} = \frac{\sin 30^\circ}{\sin 100^\circ} \]

### Step 9: Calculate the Sine Values
Using known sine values:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \sin 100^\circ = \sin(90^\circ + 10^\circ) = \cos 10^\circ \)

### Step 10: Write the Final Ratio
Thus, the ratio becomes:
\[ \frac{a}{c} = \frac{\frac{1}{2}}{\cos 10^\circ} = \frac{1}{2 \cos 10^\circ} \]

### Conclusion
The ratio of the smallest side to the greatest side of the triangle is:
\[ \frac{1}{2 \cos 10^\circ} \]