We are given a ladder leaning against a wall, making an angle of 60° with the ground, and the foot of the ladder is 3 meters from the wall. We need to find the length of the ladder.

This is a classic right-angled triangle problem, where:

• The length of the ladder is the hypotenuse $L$.
• The distance from the foot of the ladder to the wall is the adjacent side, $3 \, \text{m}$.
• The angle between the ground and the ladder is $60^\circ$.

We can use the cosine function, as it relates the adjacent side and the hypotenuse in a right triangle:

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Substituting the known values:

$\cos(60^\circ) = \frac{3}{L}$

Since $\cos(60^\circ) = 0.5$:

$0.5 = \frac{3}{L}$

Solving for $L$:

$L = \frac{3}{0.5} = 6 \, \text{m}$

Thus, the length of the ladder is $6 \, \text{meters}$.

Would you like further details or have any other questions?

Related Questions:

1. What would happen if the angle was 45° instead of 60°?
2. How do you calculate the height of the ladder on the wall in this scenario?
3. What trigonometric function relates the opposite side to the hypotenuse in this situation?
4. How can the Pythagorean theorem be used to check the ladder’s length here?
5. How does changing the distance from the wall affect the ladder's required length?

Tip:

Always use the correct trigonometric function that relates the sides you have and the one you need to find.