Question

find: \tan v

Explanation:

Step1: Recall the definition of tangent in a right triangle

In a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$ with respect to angle $\theta$.

Step2: Identify the sides relative to angle \( V \)

In right triangle \( UVT \) (right - angled at \( U \)):

• The side opposite to angle \( V \) is \( UT = 9 \)
• The side adjacent to angle \( V \) is \( UV \). First, we need to find the length of \( UV \) using the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with hypotenuse \( c \) and legs \( a,b \), \( c^{2}=a^{2}+b^{2} \). Here, hypotenuse \( VT = 41 \) and one leg \( UT = 9 \). Let \( UV=x \), then \( 41^{2}=x^{2}+9^{2} \)

\( x^{2}=41^{2}-9^{2}=(41 + 9)(41 - 9)=50\times32 = 1600 \)
So, \( x = UV=\sqrt{1600}=40 \)

Step3: Calculate \( \tan V \)

Using the definition of tangent, for angle \( V \), \( \tan V=\frac{\text{opposite to }V}{\text{adjacent to }V}=\frac{UT}{UV}=\frac{9}{40} \)

Answer:

\(\frac{9}{40}\)