Question

You are driving into st. louis, missouri, and in the distance you see the famous gateway-to-the-west arch. this monument rises to a height of 192 m. you estimate your line of sight with the top of the arch to be 7.82 ° above the horizontal. approximately how far (in kilometers) are you from the base of the arch?

Answer 1

You know from SOH CAH TOA that the tangent gives the ratio of the opposite side of a right triangle (height of the arch) to the adjacent side (distance to the arch). That is ...

... Tan(θ) = O/A

... tan(7.82°) = (0.192 km)/(distance to arch)

Multiplying by the denominator and dividing by the tangent gives ...

... distance to arch = 0.192 km/tan(7.82°) ≈ 1.398 km

You are 1.398 km from the arch.