Mathematics

Mathematics

Problem

A ladder 10 m10\,\text{m} long rests against a vertical wall. The foot of the ladder is on level ground and makes an acute angle θ\theta with the ground such that tan⁡θ=34\tan\theta = \dfrac{3}{4}. (i) Determine sin⁡θ\sin\theta and cos⁡θ\cos\theta. (ii) Find the height of the top of the ladder above the ground. (iii) Find the horizontal distance of the foot of the ladder from the wall. (iv) Verify that your answers for parts (ii) and (iii) satisfy the Pythagorean theorem for the triangle formed by the ladder, the wall and the ground.
Intuitive explanation

This problem uses a right triangle formed by the ladder, wall, and ground to practice converting between trigonometric ratios and applying them to find real distances. The key insight is that tan⁡θ=34\tan\theta = \frac{3}{4} tells you the ratio of opposite to adjacent sides, which you can use to construct a reference triangle and find sin⁡θ\sin\theta and cos⁡θ\cos\theta without a calculator. Once you have those ratios, multiply by the ladder's length (the hypotenuse) to get the actual heights and distances, then check your work using a2+b2=c2a^2 + b^2 = c^2.

Intuitive explanation

This problem uses a right triangle formed by the ladder, wall, and ground to practice converting between trigonometric ratios and applying them to find real distances. The key insight is that tan⁡θ=34\tan\theta = \frac{3}{4} tells you the ratio of opposite to adjacent sides, which you can use to construct a reference triangle and find sin⁡θ\sin\theta and cos⁡θ\cos\theta without a calculator. Once you have those ratios, multiply by the ladder's length (the hypotenuse) to get the actual heights and distances, then check your work using a2+b2=c2a^2 + b^2 = c^2.

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