In our Introductory Article (see COMPETITOR No. 1.), we showed how the Trigonometrical angle is supposed to be generated, and how any such angle exceeding the sum of four right angles, or 360°, may be reduced to the corresponding angle less than 360°.

From the definitions of the trigonometrical ratios of an angle (see COMPETITOR No. 4, p. 120), it is evident that those for any angle exceeding 360° are identical with the same ratios of the corresponding angle less than 360°, and that when the measure of the angle lies between 90° and 270°, the perpendicular from a point in one of the lines containing it to the direction of the other line, will fall upon the continuation of this other line beyond the vertex of the angle; also, when the measure of the angle lies between 180° and 360°, the perpendicular will fall in a direction the opposite of that in which it falls when the measure of the angle does not exceed 180°.

To show these different positions of the perpendicular, and consequently to fix limits between which a trigonometrical angle having a given ratio lies, the following convention has been adopted.

Of the two lines containing an acute angle, calling that one which is considered as fixed the horizontal, and from the vertex of the angle drawing a line (which we shall call the vertical), at right angles to it, then the measure of any perpendicular to either of these fixed lines (the horizontal or vertical) from any point on the side of that line on which the acute angle lies is positive, and from any point on the opposite side, negative.

The measures of any parts of the lines containing the angle are, in every position, considered positive, but of their continuations, negative.

From the convention above stated it follows:

(1.) The measure of the perpendicular from a point in one of the lines containing an angle to the other, or to its direction, is positive if the measure of the angle lies between 0 and 180°, and negative if between 180° and 360°.

(2.) The measure of the intercept between the vertex of the angle and the foot of the perpendicular is positive when the angle lies between 0 and 90°, or between 270° and 360°, and negative when it lies between 90° and 270°.

Hence all the trigonometrical ratios of an angle not exceeding 90° are positive.

When the angle lies between 90° and 180°, the foot of the perpendicular is on the continuation of the line beyond the vertex of the angle, the measure of this intercept is, therefore, negative, and as it is the only negative measure which is found in the trigonometrical functions of such an angle, every ratio, of which it forms one of the terms, is negative, viz., the cosine, secant, tangent, and cotangent, and the others, viz., the sine, and its reciprocal the cosecant positive.

(3.) When the angle lies between 180° and 270°, the measures of both perpendicular and intercept are negative; any function therefore into which only one of these enters is negative, but into which both enter is positive, the tangent and its reciprocal the cotangent are therefore posstive, and the others negative.

(4.) When the angle lies between 270° and 360°, the measure of the perpendicular is the only

negative term, and therefore any ratio which contains it is negative, viz., the sine, cosecant, tangent, and cotangent, and the remaining two, viz., the cosine and its reciprocal, the secant, are positive.

Summary showing the algebraic signs of the six principal trigonometrical ratios of an angle:

If the three principal trigonometrical ratios which have not the prefix co are taken in this order,

1. Sine.

2. Tangent.

3. Secant.

we may lay down the following rules:

(1.) If when the number of degrees in the given angle is divided by 90, the quotient is 0, the angle has no negative ratio.

(2.) If the quotient is 1, the sine and consequently its reciprocal the cosecant are positive, and the remaining four ratios negative.

(3.) If the quotient is 2, the tangent and its reciprocal the cotangent are positive, and the others negative.

(4.) If the quotient is 3, the secant and its reciprocal the cosine are positive, and the sine, cosecant, tangent, and cotangent, negative.

It has been already stated that the trigonometrical ratios of any angle are identical with those of the corresponding angle less than 360°, if therefore the measure of the given angle is not less than 360°, divide the number of degrees it contains by 360, and to the remainder apply the above rules.

Without, however, reducing the given angle to the corresponding one less than 360°, we may state the above rules thus :

(1.) If when the number of degrees in the given angle is divided by 90, the quotient is of the form 4n, where n is 0, or any whole number, 1, 2, 3, &c., all the trigonometrical ratios of the angle are positive.

(2.) If the quotient is of the form 4n+ 1, the sine, and its reciprocal, the cosecant, alone are positive.

(3.) If the quotient is of the form 4n + 2, that is 2 (n + 1), the tangent, and its reciprocal the cotangent only are positive.

(4.) If the quotient is of the form 4n + 3, the secant, and its reciprocal the cosine are the only positive ratios.

Variations and limits of the trigonometrical ratios in magnitude.

From the manner in which the trigonometrical angle is supposed to be generated (see COMPETITOR No. 1, p. 8), it is plain that the smaller the angle the smaller is the numerator of the fraction which expresses its sine, while the denominator remains unchanged, when, therefore, the angle is indefinitely

diminished, so is its sine; consequently we have

sine of 0° = 0.

Since for this value of the measure of an angle the two lines containing it coincide, the numerator and denominator of the fraction expressing its cosine, also coincide in magnitude, its cosine is therefore equal to 1.

Hence,

As the line OB revolves in the direction indicated by the positions B, B, B, &c., the numerator of the fraction expressing the sine increases till the revolving line comes into the position indicated by B, that is till the measure of the angle increases to 90°, when the numerator becomes equal to the denominator which remains the same throughout; wherefore, as the angle increases from 0 to 90°, the sine increases from 0 to 1. Therefore,

Sine of 90° = 1.

Since, as is evident from the figure, the intercept between the vertex of the angle and the foot of the perpendicular (the measure of which intercept is the numerator of the fraction expressing the cosine), diminishes as the angle increases from 0 to 90°, while for an angle of the latter magnitude it vanishes; therefore,

cosine of 90° = 0.

Hence as the sine of an angle passes through all magnitudes from 0 to 1, and the cosine from 1 to 0, while the angle passes through all magnitudes from 0 to 90°, the sine and cosine must coincide in value for an angle of 45°; also the sine of an angle between 0 and 45° is less, and between 45° and 90°, greater than its cosine.

also that as the angle increases the numerator of the fraction expressing its tangent increases and its denominator diminishes, the tangent therefore increases in a two-fold ratio, and as the measure of the angle approaches 90°, the numerator approaches the length of the revolving line, while the denominator approaches 0; therefore,

Also, as in the case of the sine and cosine, the tangent and cotangent are equal when the angle is 45°, each being then equal to 1, and when the angle lies between 0 and 45° its tangent is less, and when between 45° and 90° greater than its cotangent.

The secant, being the reciprocal of the cosine, is 1 when the cosine is 1, that is when the angle is 0, and infinity when the cosine is 0, that is when the angle is 90°; therefore,

Similarly, the cosecart, being the reciprocal of the sine, is infinity when the sine is 0, that is when the angle is 0, and 1 when the sine is 1, that is when the angle is 90°; hence,

These values for the secant and cosecant of 0 and 90° may, as in the case of the sine and cosine, be derived directly from the figure.

NOTE.-The sine, tangent, and secant of an angle increase in value as the angle increases from 0 to 90°. but the other three functions, viz., those having the prefix co diminish.

Want of space obliges us to hold over the remainder of this Article.

FEMALE CLERKS' EXAMINATION.

JANUARY, 1884.

SUBJECTS FOR ENGLISH COMPOSITION.

Time, 1 hour.

In this Exercise attention should be paid to Handwriting, Spelling, Punctuation,
Grammar, and Style.

You may choose any one of the following subjects:

1. An Account of a Family Gathering;

Or

2. Schools of Cookery;

Or,

3. Novels and Life.

The Composition should not fill less than two folio pages.

GEOGRAPHY.
Time, 1 hour.

Extra marks will be given for neatly drawn maps, but only so far as they are accurate.

1. Give a brief description of the Geography of England as regards its mountain system, its drainage, and its coast line.

2. Name and give the positions of the capitals of all the continental States of Northern Europe that border on the English Channel, the North Sea, and the Baltic. What is the form of Government of each State, and what is its principal article of export trade.

3. Describe the position of ten of the largest islands upon the west coast of Great Britain, and give a short account of one of them.

4. On the accompanying map of Southern Europe* mark the position of the different countries, the course of one principal river through each, and the names of the capes, bays, and inlets of Italy and Spain. Fix also the position of Madrid, Gibraltar, Lyons, Marseilles, Genoa, Venice, Rome, Brindisi, Trieste, Dulcigno, Salonica, Athens, Barcelona, and Sofia.

5. What possessions has Great Britain on the Continent of North America? Explain where they are, and name their chief exports.

6. What is a lake, a lough, loch, and in what parts of the kingdom are these terms used? Give two examples of each, stating where they are, and how they are fed and drained.

ENGLISH HISTORY.
Time 1 hour.

1. What was meant by Mercia, Wessex, the Danelagh, the Pale, and Bretwaldas?

2. What events led to the signing of Magna Charta? What do you know of its contents?

3. Write a life either of Warwick the King Maker or of Thomas Cromwell.

4. What do you know of Calais, Bordeaux, Dunkirk, Malta, Gibraltar, Jamaica, in their relation to English history?

5. Write a short account of the reign of Mary (Tudor).

6. Which parts of England and Scotland sided with Charles I., and which with the Parliament during the Civil War?

7. What do you know of Clive, Howe, Nelson, Collingwood, Havelock, Howard of Effingham?

* This was an outline map placed before the candidates,