11. The chord of an arc is the ratio of the straight line joining the two extre mities of the arc to the radius of the circle.

PROPOSITION.

The chord of any arc is equal to twice the sine of half the are.

Take any arc AQ, subtending at the centre of the

circle the angle ACQ = 0.

Draw the straight line CP bisecting the angle ACQ.

Join A, Q; from P let fall PM perpendicular on CA.

Since CP bisects ACQ, the vertical angle of the isosceles triangle ACQ, it bisects the base AQ at right angles.

.: AO = OQ, and the angles at O are right angles.

Again, since the triangles AOC, PMC, have the angles CMP, COA, right angles, and the angle PCM common to the two triangles, and also the side CP of the one equal to the side CA of the other, these triangles are in every respect equal.

We shall now proceed to explain the principle by which the signs of the trigonometrical quantities are regulated.

All lines measured from the point C along CA, that is, to the right, are considered positive, or have the sign +.

All lines measured from the point C along Ca, that is, in the opposite direction, to the left, are considered negative, or have the sign

All lines measured from the point Calong CB, that is, upwards, are considered positive, or have the sign+.

All lines measured from the point C along Cb, that is, in the opposite direction downwards, are considered negative, or have the sign Let us determine according to this principle, the signs of the sines and cosines of angles in the different quadrants.

CM, is reckoned from C along CA, to the right, and is

positive.

In the first quadrant, therefore, the sine and cosine are both positive.

wards, and is .. positive.

CM, is reckoned from C along Ca, to the left, and

is .. negative.

In the second quadrant, therefore, the sine is positive, and the cosine is negative.

In the third quadrant, sin. = Here P,M,

Cm, is reckoned from C along Cb, downwards, and is .. negative.

CM, is reckoned from C along Ça, to the left, and

is .. negative.

In the third quadrant, therefore, the sine and cosine are both negative.

a Ma

MO

CM, is reckoned from C along CA, to the right, and

C

Ρι

is .. positive.

In the fourth quadrant, therefore, the sinc is negative and the cosine positive.

Hence we conclude, that, the sine is positive in the first and second quadrants, and negative in the third and fourth; and the cosine is positive in the first and fourth, and negative in the second and third; or, in other words,

The sine of an angle less than 180° is positive, and the sine of an angle greater than 180° and less than 360° is negative.

The cosine of an angle less than 90° is positive, the cosine of an angle greater than 90° and less than 270° is negative, and the cosine of an angle greater than 270° and less than 360° is positive.

The signs of the sine and cosine being determined, the signs of all the other trigonometrical quantities may be at once established by referring to the relations in Table I.

Hence, it appears that when the sine and cosine have the same sign, the tangent will be positive, and when they have different signs it will be negative.

Therefore, the tangent is positive in the first and third quadrants, and negative in the second and fourth.

The same holds good for the cotangent; for

the sign of the secant is always the same with that of the cosine; and, si ce

in like manner, the sign of the cosecant is always the same with that of the sine. The versed sine is always positive, being reckoned from A always in the same direction.

it is sometimes convenient to give different signs to angles themselves. We have hitherto supposed angles of different magni

tudes to be generated by the revolution of the moveable radius CP round C in a direction from right to left; and the angles so formed have been considered positive, or affected with the sign+. If we now suppose an angle to be generated by the revolution of the radius CP' in the opposite direction, we may, upon a principle analogous to the former, consider the angle as negative, and affect it with the sign

We stal now determine the variations in the magnitude of the sine and cosine for angles of different magnitudes.

In the first quadrant:

B

Let CP, CP,, CP1,

be different positions

of the revolving radius in the first quadrant; and

When the angle becomes very small, PM becomes very small also; and when he revolving radius coincides with CA, i, e. when the angle becomes 0, then PM disappears altogether, and is = 0.

On the other nand, when the angle becomes equal to 90°, PM coincides with CB, and is equal to it.

Hence sirce, generally sin. =

PM
CT

and since, when / — 90°, FM = C

Again, it is manifest, that as the angle increases the cosine diminishes; for

CM, CM,
CA CA

7

CM, CM,
7 CA

When the angle is very small, CM is very nearly equal to CA; and when the revolving radius coincides with CA, coincides with CA and is equal to it.

i. e. when the angle is 0, then CM

On the other hand, as the angle increases, CM diminishes, and when the angle becomes equal to 90°, CM disappears altogether, and is = 0.

Let us now take different positions of the revolving radius in the second quadrant.

It is manifest, that as the angle increases the sine diminishes; for

P. B

P

As the angle goes on increasing, PM goes on diminishing; and when CP coincides with Ca, i. e. when the angle becomes equal to 180°, PM disappears altogether, and is equal 0.

Hence since generally, sin. =

PM
CA'

and since, when ◊ = 180° PM = 0;

.. sin. 180° = 0.

On the other hand, as the angle increases the cosine increases; for

and when the revolving radius coincides with Ca and the angle becomes 180°,

The negative sign is here employed, because the cosine is reckoned to the left along Ca.

Peasoning in the same manner for the third and fourth quadrants, we shall

hat, as the angle increases in the first quadrant, from 0 up to 90% The sine, being positive, increases from 0 up to 1,

The cosine, being positive, decreases from I down to 0.
́hat, as the angle increases in the second quadrant, from 90° up to 1SC",
The sine, being positive, decreases from 1 down to 0,

The cosine, being negative, increases from 0 up to — 1.
That, as the angle increases in the third quadrant, from 180° up to 270°
The sine, being negative, increases from 0 up to —1,

The cosine, being negative, decreases from -1 down to 0.
t'hat, as the angle increases in the fourth quadrant, from 270° up to 360°,
The sine, being negative, decreases from -1 down to 0,
The cosine, being positive, increases from 0 up to 1.

The variations in the magnitude of the sine and cosine being known, those of the other trigonometrical quantities may be determined by means of the relations in Table I.

The truth of this last relation may be readily illustrated, by referring to the geometrical construction; when it will be seen, that for an angle of 90', AT becomes parallel to CP; and therefore, the point T, in which the two lines meet, is at an infinite distance.

We shall next proceed to point out some important general relations, which exist between the trigonometrical functions of angles less than 90° and those of angles greater than 90°.

Draw CP, making with CA any angle PCA, which we may call ; let fall PM perpendicular from P on CA. Draw CP', making with BC the angle BCP = PCA =; and from P' let fall P'M' perpendicular on Ca

Then the angle P’CA = 90° + ". The two triangles PCM, P'CM', have the side PC of the one equal to the side PC of the other, also the angles at M and M' right angles, and the angle

P

M'

A

CPM of the one equal to the angle P'CM' of the other; .. the two triangles are in every respect equal; and

• That is, considered absolutely or independently of its sign.