A ladder 6m long leans against a vertical wall so that it makes an angle of 60^o ...

In this question, we have a ladder leaning against a wall, forming a right-angled triangle. We know the length of the ladder (the hypotenuse) and the angle between the ladder and the wall. Our goal is to find the distance between the foot of the ladder and the wall (the base of the right-angled triangle).

To solve this problem, we can use trigonometry. Specifically, we can use the cosine function. The cosine of an angle in a right-angled triangle is the ratio of the adjacent side to the hypotenuse. In this case, the adjacent side is the base of the triangle, which is the distance we want to find, and the hypotenuse is the length of the ladder (6m).

Let's denote the angle between the ladder and the wall as \(\theta\), the distance between the foot of the ladder and the wall as \(d\), and the length of the ladder as \(L\). Using the cosine function, we have:

\[\cos \theta = \frac{d}{L}\]

Now, we can plug in the given values: \(\theta = 60^\circ\) and \(L = 6m\). Using the cosine of \(60^\circ\) (which is \(\frac{1}{2}\)), we get:

\[\frac{1}{2} = \frac{d}{6}\]

To solve for \(d\), we can multiply both sides of the equation by 6:

\[d = 6 \times \frac{1}{2} = 3\]

So, the distance between the foot of the ladder and the wall is 3 meters.