What’s it all About?
Measures of Angle
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!
|Degrees & Radians|
|radian = (180 ÷ pi) degrees
or approx. = 57.295 779 513 082 320 . . . .°
1 degree = 0.017 453 292 519 943 . . . . radians
90° = 1.570 796 326 794 896 619 . . . . rad
Sine Cosine Tangent
cosine (-A) = cosine A
tangent (-A) = -tangent A
This is done automatically in the calculator.
All we offer here is the definitions of the other functions and some of the relationships between them.
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.
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
One way of showing which one we are doing is by using the prefix ‘arc’ in the inverse case. As an example
For example, the range of angles starting