A straight ladder of length $L$ leans against a vertical wall, forming an angle $\theta$ with the ground. According to the Law of Sines, what is the proper distance from the feet of the ladder to the wall?

• A. $L\cdot sin\left(\theta \right)$
• B. $\frac{L}{sin\left(\theta \right)}$
• C. $\frac{L}{cos\left(\theta \right)}$
• D. $L\cdot cos\left(\theta \right)$

Answer: (D)

Verified step by step guidance

1. Identify the right triangle formed by the ladder, the wall, and the ground. The ladder is the hypotenuse with length \(L\), the wall is vertical, and the ground is horizontal.
2. Recognize that the angle \(\theta\) is between the ladder and the ground, so the side adjacent to \(\theta\) is the distance from the foot of the ladder to the wall, and the side opposite to \(\theta\) is the height the ladder reaches on the wall.
3. Recall the definition of cosine in a right triangle: \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\).
4. Express the distance from the foot of the ladder to the wall (adjacent side) in terms of \(L\) and \(\theta\) by rearranging the cosine formula: \(\text{distance} = L \cdot \cos(\theta)\).
5. Note that the Law of Sines is not necessary here because the triangle is right-angled, and basic trigonometric ratios directly give the distance.