The ladder leaning against the wall

Statement

A ladder $5$ m long leans against a wall; its foot is $3$ m from the wall. Compute the angle the ladder makes with the ground.

Solution

The wall, the ground and the ladder form a right-angled triangle: the ladder is the hypotenuse ($5$ m) and the distance of the foot from the wall is the leg adjacent to the required angle $\theta$. Hence $\cos\theta = \frac{3}{5} = 0{,}6 \;\Rightarrow\; \theta = \arccos(0{,}6) \approx \boxed{53{,}13°}.$ (Equivalently, the height reached on the wall is $\sqrt{5^2-3^2}=4$ m, and $\sin\theta = 4/5$.)

Links

Topics: Goniometry
Concepts: Arccosine · Cosine
Functions: Arccosine · Inverse trigonometric functions
Skills: Modelling · Using formulae
Exercise types: Word problem