`Delta UVW` is right angled at. V.If sec U = 5/3 , then what is the value of tanW ?

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle
We have a right-angled triangle UVW with the right angle at V. We are given that sec U = 5/3.

### Step 2: Relate secant to the sides of the triangle
The secant of an angle in a right triangle is defined as the ratio of the hypotenuse to the adjacent side. Therefore, we can express this as:
\[ \sec U = \frac{\text{Hypotenuse}}{\text{Adjacent}} \]
Given that \(\sec U = \frac{5}{3}\), we can let:
- Hypotenuse (UW) = 5
- Adjacent side (UV) = 3

### Step 3: Use the Pythagorean theorem to find the opposite side
In a right triangle, the Pythagorean theorem states:
\[ \text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2 \]
Substituting the known values:
\[ 5^2 = 3^2 + VW^2 \]
This simplifies to:
\[ 25 = 9 + VW^2 \]
Subtracting 9 from both sides gives:
\[ VW^2 = 16 \]
Taking the square root of both sides, we find:
\[ VW = 4 \]

### Step 4: Calculate tan W
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. For angle W:
\[ \tan W = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{UV}{VW} \]
Substituting the values we found:
\[ \tan W = \frac{3}{4} \]

### Conclusion
Thus, the value of \(\tan W\) is:
\[ \boxed{\frac{3}{4}} \]

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