# 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.

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Thanks, could you please attach a sample diagram so I could better understand? Is the one below correct? – jaykirby Jun 20 '13 at 1:53
@jaykirby, Ross's picture below is exactly what I would have drawn. Is the diagram clear to you? For example, is it clear where the letters are? – Eric Kightley Jun 20 '13 at 13:58
This is a case where trying to solve a problem without first drawing the picture is almost sure to lead to an incorrect answer. – Lubin Jun 20 '13 at 21:32

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 $\arctan \frac {22.5}{1.25}-\arctan \frac {20}{1.25}=\arctan 18-\arctan 16 \approx 86.82^\circ - 86.42^\circ \approx 0.40^\circ$

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@user82306: No, you said they were 1.25m above the road. The 20 is the distance to the near edge of the letters, the 22.5 the distance to the far edge of the letters. – Ross Millikan Jun 20 '13 at 2:37
@user82306: for the first, you are looking for an angle, so need an inverse trig function. To use the sides of the triangle directly, you need a right triangle. So I have two angles at the driver's eye: one from straight down to the near edge of the letters and one from straight down to the far edge. I calculate the two angles with the arctangent and subtract to get the value of the small angle between them. – Ross Millikan Jun 20 '13 at 2:39
@user82306 arctan is another name for inverse tangent: $\arctan x = \tan^{-1} x$. They are just two ways of referring to the same function. So everywhere you see $\arctan$ in Ross's answer, you can replace it by $\tan^{-1}$. The answer will be the same as the one given by Ross. – amWhy Jun 20 '13 at 3:44
@user82306: As you said the letters were painted on the road, I took them to be horizontal. In my diagram the driver is at the vertical side of the triangle, with eyes at the top looking to the left. Do you disagree? – Ross Millikan Jun 20 '13 at 3:52
@user82306: then please draw your version or describe it better. I don't understand what you are thinking, and you haven't described it except for the original post. – Ross Millikan Jun 20 '13 at 4:10

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

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I don't quite understand, could you please elaborate why you would use 2.5-1.25? – jaykirby Jun 19 '13 at 16:36
Your driver is 1.25 meters above the road and the sign is 2.5 above the road. You have to match these heights to make a perfect triangle. – Salieri Jun 19 '13 at 16:39
I think the letters are horizontal, as they are painted on the road. – Ross Millikan Jun 19 '13 at 19:30
I think @Cardonai is correct...could you please include a diagram to show your reasoning? – jaykirby Jun 20 '13 at 2:05