`Delta`UVW is the right angled at V. If sinU=4/5, then what is the value of cosec W?

Options

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

Answer

To solve the problem, we need to find the value of cosecant W in triangle UVW, which is a right triangle with the right angle at V. We are given that sin U = 4/5.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
In triangle UVW, angle V is the right angle. Therefore, angles U and W are complementary, meaning U + W = 90 degrees.

2. **Using the Sine Function**:
We know that sin U = opposite/hypotenuse. Given sin U = 4/5, we can identify:
- Opposite side to angle U (which is side VW) = 4
- Hypotenuse (side UW) = 5

3. **Finding the Third Side Using Pythagoras Theorem**:
We can find the length of the adjacent side (side UV) using the Pythagorean theorem:
\[ UV^2 + VW^2 = UW^2 \]
Substituting the known values:
\[ UV^2 + 4^2 = 5^2 \]
\[ UV^2 + 16 = 25 \]
\[ UV^2 = 25 - 16 = 9 \]
\[ UV = \sqrt{9} = 3 \]

4. **Finding Cosecant W**:
Cosecant is the reciprocal of sine. Since W and U are complementary angles:
\[ \sin W = \cos U \]
We can find cos U using the sides we have:
\[ \cos U = \frac{adjacent}{hypotenuse} = \frac{UV}{UW} = \frac{3}{5} \]
Therefore,
\[ \sin W = \cos U = \frac{3}{5} \]
Now, cosecant W (cosec W) is the reciprocal of sin W:
\[ \csc W = \frac{1}{\sin W} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \]

### Final Answer:
The value of cosec W is \( \frac{5}{3} \).