`Delta` UVW is right angled at V. If cosU = 8/17, then what is the value of sinW ?

Options

A. `15/17`
B. `17/8`
C. `8/17`
D. `17/15`

Answer

To solve the problem step by step, we will analyze the right-angled triangle UVW, where angle V is the right angle, and we are given that cos U = 8/17. We need to find the value of sin W.

### Step 1: Understand the triangle
In triangle UVW:
- Angle V is the right angle.
- Angle U and angle W are the other two angles.

### Step 2: Use the cosine definition
We know that:
\[ \cos U = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
From the problem, we have:
\[ \cos U = \frac{8}{17} \]
This means that the length of the adjacent side (base) to angle U is 8, and the length of the hypotenuse is 17.

### Step 3: Find the length of the opposite side
To find the length of the opposite side (perpendicular) to angle U, we can use the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2 \]
Substituting the known values:
\[ 17^2 = 8^2 + \text{Opposite}^2 \]
\[ 289 = 64 + \text{Opposite}^2 \]
\[ \text{Opposite}^2 = 289 - 64 \]
\[ \text{Opposite}^2 = 225 \]
Taking the square root:
\[ \text{Opposite} = 15 \]

### Step 4: Find sin W
Now, we need to find sin W. By definition:
\[ \sin W = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
From the triangle:
- The opposite side to angle W is the same as the opposite side to angle U, which we found to be 15.
- The hypotenuse remains 17.

Thus:
\[ \sin W = \frac{15}{17} \]

### Final Answer
The value of sin W is:
\[ \sin W = \frac{15}{17} \]