# Confusing Trigonometry Problem

Lets say at an intersection the words “STOP HERE” are painted on the road in red letters 2.5m high. It is important that drivers using this lane can read the letters. How can I find the angle subtended by the letters to the eyes of a driver 20m from the base of the letters and 1.25m above the road?

Is it right to use tan, so tanθ=1.25/20? Or am I missing something?

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I find it most helpful to start any problem like this by drawing a diagram. The most useful perspective to consider is that of an observer on the sidewalk. Take a 2-dimensional cross-section of what she sees. Such a drawing will include a triangle. The bottom edge of this triangle represents the letters, so that the two vertices of this edge are the top and the bottom of the letters. The last vertex represents the driver’s eyes. You can then add the other information you have. For example, you know how far away the driver is from the lettering as measured down the road.

Once you have this picture, you can use trigonometric / geometric identities to find relevant lengths and angles. An important question to ask yourself is whether you have enough information given to you to find what you need. Are you comfortable using trigonometric identities and laws?

If this description of the picture is not clear, please let me know and I will attach a sample diagram.

As others have said, the first step is to draw a picture. The driver’s eye is at the top of the 1.25, and the letters cover the thick 2.5 along the bottom. The angle to the bottom of the “Stop Here” is $\theta = \arctan \frac {20}{1.25}$ and the angle to the top of the “Stop Here” is $\arctan \frac {22.5}{1.25}$, so the angle subtended by the letters at the driver’s eye is $\arctan \frac {22.5}{1.25}-\arctan \frac {20}{1.25}=\arctan 18-\arctan 16 \approx 86.82^\circ – 86.42^\circ \approx 0.40^\circ$

Just use the tangent (2.5-1.25)/20.