`Delta UVW` s right angled at V. If sinU = 24/25, then what is the value of cosW ?

Options

A. 25/7
B. 24/7
C. `7//24`
D. 24/25

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the triangle and the given information
We have a right-angled triangle ΔUVW with the right angle at V. We are given that sin U = 24/25.

### Step 2: Recall the definition of sine
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, for angle U:
\[ \sin U = \frac{\text{Opposite side to U}}{\text{Hypotenuse}} = \frac{VW}{UW} \]
Given that \(\sin U = \frac{24}{25}\), we can set:
- Opposite side (VW) = 24
- Hypotenuse (UW) = 25

### Step 3: Use the Pythagorean theorem to find the adjacent side
In a right triangle, the relationship between the sides is given by the Pythagorean theorem:
\[ UW^2 = UV^2 + VW^2 \]
Substituting the known values:
\[ 25^2 = UV^2 + 24^2 \]
Calculating the squares:
\[ 625 = UV^2 + 576 \]
Now, isolate UV^2:
\[ UV^2 = 625 - 576 = 49 \]
Taking the square root gives:
\[ UV = 7 \]

### Step 4: Recall the definition of cosine
Now we need to find cos W. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. Therefore, for angle W:
\[ \cos W = \frac{\text{Adjacent side to W}}{\text{Hypotenuse}} = \frac{UV}{UW} \]
Substituting the values we found:
\[ \cos W = \frac{7}{25} \]

### Final Answer
Thus, the value of \(\cos W\) is:
\[ \cos W = \frac{7}{25} \]