To find the angle that the ladder makes with the horizontal, we can consider the right-angled triangle formed by the ladder, the wall, and the ground. Here, the wall is perpendicular to the ground, and the ladder forms the hypotenuse of the triangle.
We can use trigonometric ratios to find this angle. Specifically, we will use the tangent ratio, which is the ratio of the side opposite to the angle to the side adjacent to the angle.
Let's denote:
- The angle between the ladder and the ground as θ.
- The height of the wall (opposite side to θ) as `opposite = 1.6` meters.
- The distance from the foot of the ladder to the wall (adjacent side to θ) as `adjacent = 4` meters.
The tangent of the angle θ is given by:
tan(θ) = opposite / adjacent.
We can calculate this ratio directly:
tan(θ) = 1.6 / 4.
Now, we need to find the angle θ whose tangent is 1.6 / 4. We can use an inverse trigonometric function called arctangent (commonly denoted as atan) to find this angle. However, to make this calculation, we generally need a scientific calculator or some computational tool that can compute arctangent.
Let's calculate it manually:
tan(θ) = 1.6 / 4
tan(θ) = 0.4.
So we are looking for the angle θ such that tan(θ) = 0.4. Using a calculator or table to find the arctangent of 0.4, we will find the angle in radians. Then we can convert it to degrees since angles are often more understandable when expressed in degrees.
θ ≈ atan(0.4).
Calculating this value using a scientific calculator or another method would provide us with the angle in radians, and then converting it into degrees using the relationship:
degrees = radians × (180/π).
In summary, without a calculator, we cannot give a numerical answer for the angle θ, but the method described here is how you would find the angle which the ladder makes with the horizontal. With a calculator, you would end up with a specific value for θ in degrees.
