`DeltaDEF` is right angled at E . If `cosecD=(5)/(4)` ,what is the value of cosec F?

Options

A. `(5)/(3)`
B. `(3)/(4)`
C. `(4)/(5)`
D. `(4)/(3)`

Answer

To solve the problem step by step, we will analyze the right-angled triangle DEF, where the right angle is at E, and we are given that cosec D = 5/4. We need to find the value of cosec F.

### Step-by-Step Solution:

1. **Understanding Cosecant**:
We know that cosec D = 1/sin D. Given that cosec D = 5/4, we can find sin D.
\[ \sin D = \frac{1}{\text{cosec D}} = \frac{4}{5} \]

**Hint**: Remember that cosecant is the reciprocal of sine.

2. **Assigning Triangle Sides**:
In a right triangle, we can assign the lengths of the sides based on the sine value. If we let the opposite side to angle D (perpendicular) be 4x and the hypotenuse be 5x, we can represent the triangle as follows:
- Opposite side (DE) = 4x
- Hypotenuse (DF) = 5x

**Hint**: The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.

3. **Finding the Adjacent Side**:
To find the length of the adjacent side (EF), we can use the Pythagorean theorem:
\[ DF^2 = DE^2 + EF^2 \]
Substituting the known values:
\[ (5x)^2 = (4x)^2 + EF^2 \]
\[ 25x^2 = 16x^2 + EF^2 \]
\[ EF^2 = 25x^2 - 16x^2 = 9x^2 \]
Taking the square root:
\[ EF = 3x \]

**Hint**: The Pythagorean theorem relates the lengths of the sides in a right triangle.

4. **Finding Cosecant of Angle F**:
Now that we have the lengths of the sides, we can find cosec F. Cosec F is defined as:
\[ \text{cosec F} = \frac{\text{Hypotenuse}}{\text{Opposite to F}} \]
Here, the hypotenuse (DF) is 5x and the opposite side to angle F (DE) is 3x. Therefore:
\[ \text{cosec F} = \frac{5x}{3x} = \frac{5}{3} \]

**Hint**: Cosecant is the ratio of the hypotenuse to the opposite side of the angle.

5. **Final Answer**:
The value of cosec F is:
\[ \text{cosec F} = \frac{5}{3} \]