Measures of Angle

Everyone who has had any sort of formal education at all will have learned about angles. How to measure them and how to draw them. They will certainly know about degrees. Most, but not all, will have gone on to do trigonometry with right-angled triangles, and words like sine cosine tangent will have some (half-remembered?) meaning for them. Some will have gone on to apply that trigonometry to any triangle which requires working with angles beyond 90 degrees and the manipulation of more complex formulas. And that will be it for the great majority.

Only a few would have gone further into the more arcane knowledge of angles. This means measurement expressed in radians, and the range of ratios (probably now known as functions) would have been expanded to include cosecant secant cotangent and even a few others. More relationships and formulas would have been encountered. All of this would be done either as a logical extension of mathematics or to satisfy some professional requirements. The purpose of these notes then, is not to teach anything about angle measures, but to provide help for those who have some sort of previous acquaintance with the subject.

There are three.
The best known is undoubtedly the DEGREE which is very simply defined as 1/360th part of a full-turn. It was so defined by the Babylonians over 4000 years ago and is the oldest basic unit of measure which has not changed its size since it was first used.
It can be sub-divided into 60 minutes, and each of those into 60 seconds.
One important measure is the right-angle which is 90° and equal to one-quarter of a complete turn.

There is also the GRADE which is similiar to the degree, but divides the full-turn into 400 parts, putting 100 grades in a right-right. So, 100 grades = 90 degrees.
A centigrade is 1/100th of a grade and a measure of angle, which is why it was ruled in 1947 that it should NOT be used as a measure of temperature! The grade is little used now, but is still found marked on angle-measuring instruments and calculators.

The RADIAN is the SI standard unit of angle measurement and is the unit used in all serious mathematical work. To define it, draw a circle (of any size) and mark off an arc on its circumference which is equal in length to the radius. (This is not possible in practice, but it is a theoretical construction.) Draw straight lines from the two ends of that arc to the centre of the circle. The angle between them is 1 radian.
It is this dependence on a circle which invokes pi when a connection is made between radians and degrees.
The name was devised in the 1870’s but there is some controversy over who was the first to use it. It is a contraction of ‘radial angle’ Its abbreviation is rad and not r as is sometimes seen.
A millirad is 1/1000th of a radian and about 0.57 of a degree or 3.44 minutes. It is also known as a mil and used in directing and correcting the aim of guns BUT, in that case, definitions of the mil vary according to which army is doing the shooting!

For many the above three words, together with a right-angled triangle, will always invoke a memory of trigonometry, and it may be their only memory! A phrase such “Sine equals opposite over hypotenuse” might have lingered or even the mystical incantation “SOHCAHTOA” which is possibly remembered with the aid of a suitable sentence like “Should Old Harry Catch Any Herrings Trawling Off America?” Whether any of it is still ‘useable’ is another matter. Each of those three words produces a number which gives the ratio between two particular edges of a right-angled triangle. The edges are usually identified as ‘opposite’, ‘adjacent’ and ‘hypotenuse’. Definitions will not be offered here.

Knowing the value of that ratio and the length of just one edge enables the length of any other edge to be worked out. In the past it was necessary to have a set of tables giving the values of the ratios for any size of angle, but the ready availability of scientific calculators has removed the need for those. Another help which is not apparent is that the angles could not be bigger than 90 degrees and thus their ratio must always be positive and unambiguous. Neither of those words has to be true when the move is made to work with any other sort of triangle, usually referred to as the ‘general case’.

Mainly angles are thought of as being positive, if only because they must exist in order to be measured! However, there is a need to be able to work with negative angles as well and that can mean knowing how to find the sine, cosine and tangent values of those negative angles. The three expressions on the right show how, for any negative angle -A it is only necessary to find the value of the ratio for positive A and then put a negative sign in front of that value (or not) as the case might be

sine (-A) = -sine A
cosine (-A) = cosine A
tangent (-A) = -tangent A
This is done automatically in the calculator.

Previously, sin, cos and tan were referred to as ratios because of the way they were used, as outlined. But, in the wider context of mathematics they are known as functions. For an explanation of that, look elsewhere.
All we offer here is the definitions of the other functions and some of the relationships between them.

cosecant A (cosec A) = 1 / sine A
secant A (sec A) = 1 / cosine A
cotangent A (cot A) = 1 / tangent A
coversine A (covers A) = 1 – sine A
versine A (vers A) = 1 – cosine A
haversine A (hav A) = (versine A) / 2
exsecant A (exsec A) = secant A – 1
The last is now obsolete and included only for completeness.

It is only ever necessary to know the value of the sine because all the others can be worked out from that. To do this we need

and also

Most of these functions have some limits on their values.
The sine and cosine can only be from -1 to 1
The tangent and cotangent have NO limits.
The cosecant and secant have NO upper or lower limits, but cannot be between -1 and 1
The coversine and versine can only be from 0 to 2
The haversine can only be from 0 to 1

Reminders on these limits are given in the calculator.

The three versine functions (which are all positive) were designed with the aim of making calculations easier by removing the need to work with negative numbers. An example of this can be seen with the cosine rule. Conventionally it is

but its versine equivalent is

(b ~ c) means to take the smaller from the larger. Now, whatever the size of angle A, no negative numbers have to be handled. Of course it means that the versine values have to be available somewhere.

There are usually just one of two things we wish to do. Either we want to change an angle-size into a function-value OR we want to change a function-value into an angle-size. The second of these is known as the inverse.
One way of showing which one we are doing is by using the prefix ‘arc’ in the inverse case. As an example

sin 30° = 0.5 OR arcsine 0.5 = 30°

This seems clear enough but there is a problem. For any size of angle there is only ever one value to match it in each of the functions. In other words, angle to function-value is unambiguous. Unfortunately, it is not true for the inverse case.
For example, the range of angles starting

30° 150° 390° 510° 750° . . . all have a sine value of 0.5