If `Delta DEF` is right angled at E, DE = 15 and `angle DFE = 60^(@),` then what is the value of EF ?

Options

A. `5 sqrt3`
B. 5
C. 15
D. 30

Answer

To find the value of EF in the right-angled triangle DEF, where DE = 15 and angle DFE = 60°, we can use trigonometric ratios.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Values**:
- We have a right-angled triangle DEF with the right angle at E.
- DE = 15 (the side opposite to angle DFE).
- Angle DFE = 60°.

2. **Use the Sine Function**:
- In triangle DEF, we can use the sine function to find EF.
- The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
- Here, we can write:
\[ \sin(60°) = \frac{DE}{DF} \]
- Since DE = 15, we have:
\[ \sin(60°) = \frac{15}{DF} \]

3. **Calculate DF**:
- We know that \(\sin(60°) = \frac{\sqrt{3}}{2}\).
- Therefore, we can rearrange the equation:
\[ DF = \frac{15}{\sin(60°)} = \frac{15}{\frac{\sqrt{3}}{2}} = 15 \cdot \frac{2}{\sqrt{3}} = \frac{30}{\sqrt{3}} = 10\sqrt{3} \]

4. **Use the Cosine Function to Find EF**:
- Now we can use the cosine function to find EF:
\[ \cos(60°) = \frac{EF}{DF} \]
- Since \(\cos(60°) = \frac{1}{2}\), we can write:
\[ \frac{1}{2} = \frac{EF}{10\sqrt{3}} \]

5. **Calculate EF**:
- Rearranging gives:
\[ EF = \frac{1}{2} \cdot 10\sqrt{3} = 5\sqrt{3} \]

### Final Answer:
The value of EF is \(5\sqrt{3}\).