Extension of trigonometric functions to arbitrary angles

sin

As a result of the linkage between the definition of the trigonometric function and right-angled triangles, it was only possible to deal with sharp angles, which is obviously an extraordinary restriction. Here again mathematics has managed to free itself of this restriction. In the earlier discussion of each of these functions, we have presented a definition which totally corresponded to the original one. For sin it was:

The sin of an angle equals the length
of the corresponding height in the unit circle.

On the basis of this definition, we have already gained the extensions of angles to 0º and 90º, which for a in a right-angled triangle are impossible.

Viewing OA as starting leg and OP as movable leg of an angle, a continuation of the rotation to arbitrary angles suggests itself and to define sin a as the measure of the corresponding height in the unit circle. Since you are in an oriented plane, you must allot to the heights the leading signs of the coordinate system.

By the figure above, for example, sin a₁ is the length of P₁Q₁, whence it is positive, sin a₂ the length of P₂Q₂, whence it is negative, etc. If you fix the starting leg OA, the moving leg passes through the quadrants of the coordinate system, whence you talk of angles in the quadrants I, II, etc. and define the leading signs as follows:

sin of the angles in quadrants I and II is positive,
in quadrants III and IV negative.

For 0º, PQ = 0, whence sin 0º = 0, for 90º, PQ = 1, whence sin 90º = 1, for 180º PQ = 0, whence sin 180º = 0. Similarly, you find that sin 270º = -1, sin 360º = 0, and if you continue to rotate the movable leg, sin 450º = 1, sin 540º = 0, etc. For negative angles, you find sin (-90º) = -1, sin (-180º) = 0 etc.

The values of sin lie between +1 and -1.

The scale of a is arbitrary. If you use a larger scale, the curve is stretched, for a smaller scale, it is steeper. In order to obtain the so called normal curve, you should use the same scale on the two axes. It is then assumed that in the functional equation y = sin x the variables y and x use the same unit. y is a pure number, whence you should also reduce the angle a to a pure number, that is, to its arc. Then the number +1 is represented on both axes in terms of the unit radius. As you roll the unit circle along the abscissa, it gives you the ordinates of the normal curve.In order to save effort, it is worthwhile to make a pattern of the normal curve.

Since sin a lies between +1 and -1, the curve must lie between two parallels which lie at distances ±1 from the abscissa. The graph displays also all peculiarities of the function, for example, that sin a is positive in the quadrants I and II, negative in the quadrants III and IV, that sin 0º = 0, sin 90º = 1, sin 180º = 0, sin 270º = -1, sin 360º = 0, sin 30º = /2, sin 45º = /2, sin 60º = /2, etc.

The symmetry of the circle is reflected in the curve of the function

The points P₁, P₂, P₃, P₄ of the circle, which lie symmetrically with respect to the coordinate axes, allow conclusions regarding the points P'₁, P'₂, P'₃, P'₄ of the curve. At the peak of the wave AB is an axis of symmetry, at the trough of the wave CD. The wave peak and trough can be brought to cover one another by a rotation by 180º about the centre E or by a reflection and displacement. The peak and trough of the wave are congruent with respect to each other, irrespectively of whether you let O and E or O and F correspond to each other. These properties extend through the entire range of the curve, that is beyond 0º and 360º.

cos

Corresponding to the conclusions regarding sin, you now can define:

The cos of an angle is equal to the length
of the corresponding projection in the unit circle.

cos a₁ equals the length of OQ₁ in the third last figure, that is, it is negative, cos a₂, the length of OQ₂. whence it is again negative, etc.

The cosine is positive in the quadrants I and IV, negative in I and III.

For 0º, OQ = OA, that is equal to the radius, whence cos 0º=1, for 90º, OQ=0, that is,
cos 90º=0, similarly cos 180º=-1, cos 270º=0,
cos 360º = 1, etc.

The cosine lies between +1 and -1.

The graph of cos a is readily obtained like that for sin a by using again the radius as unit. You can pick up the segments OQ and transfer them.

The curve lies again between two parallels to the abscissa.

If you compare the sin- and cos- curves, you guess that curves become each other by axial reflection, for example, in the axis AB or by parallel displacement .

If you reflect the triangle PQO in the 45º axis, it becomes the triangle P'Q'O. You have that QO = Q'P' =QR, the measure of QP = sin (45º + a), of QR = cos (45º - a), whence sin(45º+a)=cos(45 º-a). Since also RP' = QO, also sin (45º - a) = cos (45º + a). The 45º axis AB is the axis of symmetry of the sin and cos curves. Similarly, the 225º axis is a symmetry axis, etc. The cos curve has therefore all the properties of symmetry of the sin curve. It follows then that the values of cos can be obtained from the sin curve by parallel displacement along the abscissa, for example, by the 90º segment.

tan

tan of an angle equals the measure of the corresponding
segment of the tangent to the unit circle.

In order to find in the II. and III. quadrants tangent segments, the free leg of the angle must be extended beyond the vertex. Like all ordinates, tangent segments above the abscissa are positive, below the abscissa negative. For example, if tan a is equal to the length of AT, that is positive, tan a equals the length of AT₁, that is, it is negative, etc.

The tan function is positive in the quadrants I and III,
negative in the quadrants II and IV.

For a = 0º, AT = 0, whence tan aº =0. As a approaches 90º, tan exceeds all bounds. At 90º, the free leg is parallel to the tangent,whence there is no cut-off of the tangent, that is tan 90º has no value. You write tan a = . Moreover, tan 180º = 0, tan 270º = , etc.

The tan function assumes every value.

For the graphing of tan a, you use again the radius as unit and transfer the tangent segment AT. The angles a and 180 + a, as you can see from the left figure above, have always the same tangent segment AT. The branches of tan become each other by parallel displacement. In addition, the function has central symmetry.

For example, the angles a and - lead to opposite, but equal values AT and AT₁, whence the starting point is the symmetry point. Similarly, the angles a and 180º - a lead to opposite, but equal values AT and AT₁, whence also the point 90º is a centre of symmetry, etc.

You will have noted that the tan function consists of branches and is not continuous. On the sin and cos curves, it is always possible to advance along it from one point to another point. Such curves are said to be continuous. This is not true for the tan curve. It is at the points 90 or 270, etc. discontinuous, while the individual branches are steady.

cot

cot a equals the measure of the corresponding cotangent
segment to the unit circle.

For angles in the quadrants III and IV, the movable legs of the angle must again be extended, in order to obtain cotangent segments, Their signs correspond to those of the abscissa. For example, cot a equals the length of BT, that is, it is positive, cot a equals the length of BT₁, whence it is negative.

cot is positive for angles in the quadrants I and III,
negative in the quadrants II and IV.

As a tends to 0, cot a grows beyond all limits. For a = 0, there exists no intersection with the cotangent and you write cot 0º = . cot 90º = 0, cot 180º = , cot 270º = 0

The cot function assumes every value.

The graph is obtained in a similar manner as the graph of tan. The Points of discontinuity are at 0º, 180º, etc. The individual branches can be shifted into each other; there is also again axial symmetry about the same centres as for tan.

The tan and cot curves have axial symmetry.

You can prove it by means of the left figure above, in which OR is the axis of symmetry. If you select two angles a which are symmetric with respect to OR, then also the symmetric segments BT₁ and AT are equal,whence

cot (45º + a) = tan (45º - a).

This statement applies to all values of a. The above figure of tan and cot shows that the ordinate at 45º is an axis of symmetry. Since the branches can be obtained from each other by parallel displacement, you obtain the proofs for other axes of symmetry.

Alternative formulation of the definitions. Proportional numbers

In the above definitions, we started from the unit circle and adopted the radius as measuring unit. sin is then equal to the measure of the corresponding height, that is, it indicates how often the unit or radius is contained in the height. If you employ another unit, the definition must be

sin a = y/r.

There will arise no contradictions, since all circles are similar, whence equally located segments are always in the same proportion. Furthermore, the ray theorem shows that

sin a = y₁/r₁ = y₂/r₂ = y₃/r₃ = ···.

Correspondingly, for an arbitrary radius and unit

cos a = x/r.

r is the radius vector which, in order to follow the process and achieve the sign convention, you must treat as an absolute quantity.

tan a is equal to the corresponding segment of the tangent to the unit circle, whence it tells how often the unit, that is the radius, is contained in the tangent segment. Again, by the ray theorem,

tan a = A₁T₁/r₁ = A₂T₂/r₂ = ···

or, if you take the height to the fixed leg (or its extension), with the leading sign taken into consideration,

tan a = A₁T₁/r₁ = y₁/x₁ = y₂/x₂ = y₃/x₃ = ···, that is, in general, tan a = y/x.

Similarly, cot a= y/x..

The earlier definitions can now be reformulated. If the starting leg of an angle is the abscissa and you draw from an arbitrary point of the movable leg the height to the abscissa, then

With this formulation of the definitions, it becomes clear that the values of the functions are proportional quantities, that is pure numbers. You may not even with the unit circle commit the error of setting a trigonometric function equal to a segment: It is always equal to the measure of the corresponding segment. During the history of these functions, this has been frequently confused, that is, the belief of the function being a segment. It is the achievement of Euler to have not only clarified this point, but altogether the behaviour of the functions in the different quadrants. The explanation of the functions by ratios of sides was only elaborated after Euler.

Law of permanence

The extensions of the definitions must be such that the old laws as well as possible apply and no contradictions arise (Law of permanence). Consider several of the former formulae.

Let a be an arbitrary angle in the unit circle. Then the measure of PQ = sin a, of OQ = cos a, whence sin² a+ cos² a= 1. The measure of AT is tan a. Also, as regards leading signs, always

AT/OA(=1) = QP/OQ or tan a = sin a/cos a, etc.

All of the formulae derived earlier remain valid.

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