### What’s it all 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.

**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.

### 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) degreesor approx. = 57 .295 779 513 082 320 . . . .°57.3 will usually do.1 degree = 0.017 453 292 519 943 . . . . radians90° = 1 .570 796 326 794 896 619 . . . . rad |

### Sine Cosine Tangent

**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.

### Negative Angles

cosine (-A) = cosine A

tangent (-A) = -tangent A

This is done automatically in the calculator.

### Other Functions

**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.

**cosine A = square root(1 – sine� A)**

**tangent A = sine A / cosine A**

**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

**a**

^{2}= b^{2}+ c^{2}– 2bc cos A**a**

^{2}= (b ~ c)^{2}+ 2bc vers A*(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.

### Inverses

**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°**

For example, the range of angles starting

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