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Word problems relating ladder in trigonometry

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Ladder Word Problems in Trigonometry

Ladder word problems model the ladder as the hypotenuse of a right triangle, with the wall giving the height and the ground giving the distance from the wall. Learn to set up the triangle and solve for height or distance using sine and cosine, with a worked example.

Setting up a ladder word problem

Ladder problems are classic right-triangle applications: the ladder is the hypotenuse, the wall gives the height (opposite side), and the ground gives the distance from the wall (adjacent side). The angle is measured between the ladder and the ground.

Ladder word problem A ladder 6 meters long leans against a wall, making a 65 degree angle with the ground. Using sine, the height reached on the wall is about 5.44 meters. Using cosine, the base of the ladder is about 2.54 meters from the wall. 65° height ≈ 5.44 m base ≈ 2.54 m ladder = 6 m
A 6-meter ladder at 65° reaches about 5.44 m up the wall and sits about 2.54 m from its base.

A 6-meter ladder leans against a wall, making a 65° angle with the ground. How high does it reach, and how far is its base from the wall?

Height (SOH): sin 65° = height / 6, so height = 6 · sin 65° ≈ 5.44 m.

Base distance (CAH): cos 65° = base / 6, so base = 6 · cos 65° ≈ 2.54 m.

The ladder is the longest side of the triangle because it stretches from the ground to a point on the wall — it's always opposite the right angle, which sits where the wall meets the ground. This setup is the same one used in guy wire problems and other angle word problems, just with a different real-world object playing the role of the hypotenuse.

If a problem instead gives the height and the base, use inverse tangent to find the angle: θ = tan⁻¹(height / base). This confirms whether a ladder is at a safe climbing angle, typically between 65° and 75°.

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