Background Notes on Measures
Angles
 Headings What's it all About?Angle MeasuresSin, cos, tanNegative AnglesOther FunctionsInversesTable of Values
 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 degreesMost, 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.
 For those needing to deal with actually solving a particular triangle, there are two calculators available dedicated to this purpose. They are linked from here.One is for Right-angled Triangles the other for Any Triangle
Measures of Angle
 Degrees & Radians 1 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
 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'.
Negative Angles
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.
Other Functions
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
cosine A = square root(1 - sine² A)
and also
tangent A = sine A / cosine A

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 isa² = b² + c² - 2bc cos Abut its versine equivalent isa² = (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
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
Which is fine if that is what we want, but suppose we wish to find arcsine 0.5 and have no idea what size of angle we are looking for. (It might even be negative!)
A study of the Table of Values (below) might help in seeing how, in any column the function-values repeat at regular intervals.

To deal with this situation it is important to realise that the function values are cyclic. That is, they repeat in a fixed and regular way, so that it is only ever necessary to know the size of angle which matches one function-value to be able to produce all the other possible angles (or as many as are required).
This one size is known as the principal angle and is usually the smallest possible in the range -90° to 90° or 0° to 180° depending on the function being used. It is this principal size of an angle which is given on a calculator.
There are formulas for working out all possible sizes of angle depending upon which function is being used and what the principal value is.
The relevant formulas for the three main functions are
 For the sine: A = 180n + (-1)nP For the cosine: A = 360n ± P For the tangent: A = 180n + P
where
A is any Angle;     P is the Principal angle;     n is any whole number.

Table of Values

00010inf1inf100

+++++++++
90pi/210inf1inf0010.5

+--+--+++
180pi0-10inf-1inf121

--+--++++
2703pi/2-10inf-1inf0210.5

-+--+-+++
3602pi010inf1inf100

+++++++++
4505pi/210inf1inf0010.5

+--+--+++
5403pi0-10inf-1inf121

--+--++++
6307pi/2-10inf-1inf0210.5

-+--+-+++
7204pi010inf1inf100
 inf = infinityThe table shows not only the values of the various functions for certain sizes of angle, but also whether the value is positive(+) or negative(-) in the interval between those specific values. The patterns are clear, and the table may easily be extended forward or backward (for negative angles).

Go to the top OR the ANGLES CALCULATOR

 Go to the Dictionary of Units OR the Calculator Menu