Understanding the Ladder and House Problem

This problem involves a ladder leaning against a house, which forms a right-angled triangle. The ground, the side of the house, and the ladder are the three sides of this triangle. We are given the angle of elevation of the ladder and the distance from the foot of the ladder to the house. We need to find the length of the ladder.

Identifying the Components of the Right Triangle

Let's visualize the situation:

• The ladder is the hypotenuse of the right-angled triangle.
• The distance from the foot of the ladder to the house is one of the legs of the right triangle, specifically the side adjacent to the angle of elevation.
• The height the ladder reaches on the house is the other leg, the side opposite to the angle of elevation.
• The angle of elevation is the angle between the ground and the ladder.

In this problem:

• Angle of elevation = $60^\circ$
• Distance from foot of ladder to house (Adjacent side) = $6.5$ metres
• Length of the ladder (Hypotenuse) = $L$ (unknown)

Applying Trigonometric Ratios

We need a trigonometric ratio that relates the angle of elevation, the adjacent side, and the hypotenuse. The cosine function relates these three components:

$\cos(\text{angle}) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$

Solving for the Length of the Ladder

Let's plug in the given values into the cosine formula:

$\cos(60^\circ) = \frac{6.5 \text{ metres}}{L}$

We know the value of $\cos(60^\circ)$ is $\frac{1}{2}$. So, we can write the equation as:

$\frac{1}{2} = \frac{6.5}{L}$

To find $L$, we can cross-multiply:

$1 \times L = 2 \times 6.5$

$L = 13$

Therefore, the length of the ladder is 13 metres.

Summary of Calculation

Using the cosine function, we related the angle of elevation ($60^\circ$), the adjacent side ($6.5$ m), and the hypotenuse ($L$).

$\cos(60^\circ) = \frac{6.5}{L}$

Since $\cos(60^\circ) = 0.5$ (or $\frac{1}{2}$), we have:

$0.5 = \frac{6.5}{L}$

$L = \frac{6.5}{0.5}$

$L = 13$

The length of the ladder is 13 metres.

Revision Table: Trigonometric Ratios

Here is a quick look at the basic trigonometric ratios for a right-angled triangle with angle $\theta$:

• Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Additional Information: Special Angles

It's useful to remember the trigonometric values for common angles like $30^\circ$, $45^\circ$, and $60^\circ$.

In this problem, knowing that $\cos(60^\circ) = \frac{1}{2}$ was key to a quick solution.