Trigonometry/Conventions

Conventions

There are numerous small conventions in mathematics, such as often using $\displaystyle \theta$ for an angle or using $\displaystyle x^2$ to indicate the square of $\displaystyle x$. Some of the conventions such as that a superscript ² means 'squared' are always used, or at least very nearly always used. Some conventions are more temporary.

• We tend to use $\displaystyle \alpha$, $\displaystyle \beta$, $\displaystyle \gamma$ and $\displaystyle \delta$ for angles rather than for lengths.
• It is a common custom to use $\displaystyle d$ for a distance, $\displaystyle h$ for a height, $\displaystyle t$ for a time. If we need more than one time, then as well as $\displaystyle t$ we will typically use $\displaystyle s$ too, provided we're not already using it for something else!
• We'll use $\displaystyle r$ for the radius of a circle. If we have two circles, say a large one and a small one, we might use $\displaystyle r$ and $\displaystyle R$, or if several circles we might use $\displaystyle r_1$, $\displaystyle r_2$ and $\displaystyle r_3$.

The problem with the customs and conventions is that they are not usually spelled out. You get used to them by seeing them used. Most of the time the conventions help a little with keeping track of what various symbols are being used for. Sometimes though we just have to be alert to the reuse or special use of symbols. Not all Greek letters are suitable letters for angles. $\displaystyle \pi=3.14159..$ for example is already dedicated to use as that important constant.

The real problems start with contradictory conventions. $\displaystyle \delta x$ can for example mean a small change in $\displaystyle x$ rather than the quantity $\displaystyle \delta$ times the quantity $\displaystyle x$. Sometimes you may see $\displaystyle \phi$ being used for an arbitrary angle, just like $\displaystyle \theta$, but at other times $\displaystyle \phi$ can have the special meaning of 'the golden ratio', that number for which $\displaystyle x = \frac{1}{x} + 1$

Other Letters and Notation

d = run
Δh = rise
l = slope length
α = angle of inclination

The way we label edges in a triangle may change a bit from one problem to another. There is no rule saying that we always must use c (or h) for the hypotenuse, for example. The next exercise is practice in not letting changes in notation get in the way of the mathematics.

To keep you on your toes, the symbol $\displaystyle \Delta$ as well as meaning 'triangle' is also sometimes used to indicate a change or a difference. In the next exercise it is used that way to indicate a change in the height of a road.

The diagram on the right shows some quantities relating to the slope of a road:

• d = run.
• Δh = rise, the change in height.
• l = slope length.
• $\alpha$ = angle of inclination.

Here l is the hypotenuse of a right-triangle.

You should be able to apply the Pythagorean Theorem to this diagram and quickly see that:

$\displaystyle l^2=d^2+(\Delta h)^2$

In an earlier exercise we saw roadsigns from Tanland. They showed ratios which were $\displaystyle\Delta h:d$.

As the angle of inclination $\displaystyle\alpha$ increases, so does the value of $\displaystyle{\Delta h}/d$

In the next exercise you are to use Pythagoras, but you are also to use percentages.

Slopes in Landofsine Do you remember these roadsigns from an earlier exercise? 3:5 5:8 8:13 1:10 1:1 You could convert the slopes to percentages, e.g. 8:13 would be 61.5% because $\displaystyle 8/13 \times 100% \approx 61.5%$. That's how slopes in most parts of the world are calculated - though I'd be surprised if usable roads actually get that steep. You're going to do a slightly different calculation. The mythical mountainous Landofsine is a country neigbouring on the country of Tanland. The roadsigns in Landofsine show slope as percentages. Most slope roadsigns in the world express $\displaystyle\Delta h/d$ as a percentage. However, in Landofsine they choose a different convention. They instead show $\displaystyle\Delta h/l$ as a percentage. Because $\displaystyle l$, being the hypotenuse, is always greater than $\displaystyle d$ it makes the slopes sound a little less steep, and this, they hope, will boost tourism. Your task: The Tanland roadsigns slopes of 3:5, 5:8, 8:13, 1:10 and 1:1 from that earlier exercise need to be converted to Landofsine percentages. Here is an example, 3:5 in most of the world would be a $\displaystyle 3/5 \times 100% = 60%$ slope. However we've divided the 'rise' over the 'run'. For Landofsine roadsigns we need to divide by the hypotenuse. Calculating it: $\displaystyle \sqrt{3^2+5^2}=\sqrt{9+25}=\sqrt{34}$. The percentage for the slope is now $\displaystyle 3/\sqrt{34} \times 100% \approx 51%$ Do the same calculation for the other roadsigns. The first three values using Landofsine calculation of slope should all be fairly close to $\displaystyle 51%$