Let's analyze this problem.

We are given a right triangle with:

• An angle of $69^\circ$
• An opposite side length of $5.8$
• A hypotenuse labeled as $v$

Part a: Identifying the Correct Equation

We have two possible equations provided:

1. Equation A: $\sin 69^\circ = \frac{5.8}{v}$
2. Equation B: $\sin 69^\circ = \frac{v}{5.8}$

To find the correct equation, recall that: $\sin(\text{angle}) = \frac{\text{opposite side}}{\text{hypotenuse}}$ In this case, the opposite side is $5.8$ and the hypotenuse is $v$. Therefore: $\sin 69^\circ = \frac{5.8}{v}$ This matches Equation A, so Equation A is correct.

Part b: Solving for $v$

Using Equation A, we have: $\sin 69^\circ = \frac{5.8}{v}$ To solve for $v$: $v = \frac{5.8}{\sin 69^\circ}$ Calculating this with $\sin 69^\circ \approx 0.9336$: $v = \frac{5.8}{0.9336} \approx 6.2$ So, the length of $v$ is approximately 6.2 meters, to 1 decimal place.

Would you like further details or have any questions?

Related Questions:

1. How would you solve for $v$ if the angle was different?
2. How can we use cosine or tangent in similar right-triangle problems?
3. What happens if we switch the opposite and adjacent sides in the trigonometric ratio?
4. How can you verify the answer using other trigonometric properties?
5. Why is it important to identify the opposite and hypotenuse correctly in trigonometry?

Tip:

In trigonometry, always identify each side relative to the given angle before choosing the appropriate trigonometric function.