If OP and OQ are unit radii which make a angles with the x axis of B and A respectively. Then the coordinates of P are (Cos B ; sin B) and for Q (cos A ; sin A). By inspection the angle POQ is of magnitude A - B.
Using the Pythagora theorem,
The equation has now been reduced to one of the standard forms whose solution is known. Hence a value for can be found and as the value of is known ca be calculated. For real solutions it is necessary for the value of c to be less than
A second method of solution is to use the half angle formulae ( Equations (43) and (46)
This quadratic gives two values for t from which general value of can be found.
The Inverse Notation
If sin = x where x is a given quantity numerically less than unity, wwe know that can be any one of a whole series of angles. Thus if can have a number of values. The inverse notation is used to denote the angle whose sine is x and the numerically smallest angle satisfying the relationship is chosen as the principle value. Here and in what follows we shall deal only with principle values and the statement to mean that is the angle that lies between radians whose sine is x. The statement means that is the inverse sine of x. On the continent this is sometimes written as
The graph of is, on thus that part of the graph with the x-axis horizontal and the axis vertical.As shown:-
In a similar way will be taken to denote the smallest angle whose cosine takes the same value for negative as for positive angles and we require a notation which gives an unique value of when x is given, we conventionally take as the angle lying between 0 and radians whose cosine is x.
These relationships will be found useful in some situations.
NOTE care must be taken avoid confusion between the inverse sine, cosine etc and the reciprocal of sin x, cos x etc. The latter should always be written as :-