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

The cosine, being positive, decreases from I down to 0.
That, as the angle increases in the second quadrant, from 90° up to 180o,
The sine, being positive, decreases from 1 down to 0,

The cosine, being negative, increases * from 0 up to

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

The cosine, being negative, decreases * from I down to 0.
That, as the angle increases in the fourth quadrant, from 270° up to 360o,
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 = 4; 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 P'C of the other, also

the angles at M and M' right angles, and the angle

P

M'

A

C

M

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.

an important proposition, which enunciated in words is, The sine of an angle is equal to the sine of its supplement.

that is, The cosine of an angle and the cosine of its supplement are equal in absolute magnitude, but have opposite signs.

If, as in the annexed figure, we draw CP', making

with Ca an angle aCP' equal to the angle 4, we shall find, in like manner,

P

If we draw CP', making with Cb an angle bCP' = 4, then

as is evident from the Def. 7, and the rule for signs; and, in like manner, we may proceed for angles in the fourth quadrant.

These relations being established between the sines and cosines, the corresponding relations between the other trigonometrical functions, may be deduced immediately from Table I. Thus,

tan. (90° + 4) =

sin. (90° + )
cos. (90°)

The student may exercise himself, by verifying such of the results in the following table as have not been formally demonstrated.

The results in the above table which are most frequently used, are marked with an asterisk, and ought to be committed to memory.

We have in the preceding pages confined ourselves to the consideration of angles not greater than 360°, but the student can find no difficulty in applying the above principles to angles of any magnitude whatsoever.

We shall conclude this introductory chapter, by demonstrating two propositions which are of the highest importance in our subsequent investigations. The first is,

In any right-angled triangle, the ratio which the side opposite to one of the acute angles bears to the hypotenuse, is the sine of that angle; the ratio which the side adjacent to one of the acute angles bears to the hypotenuse, is the cosine of that angle; and the ratio which the side opposite to one of the acute angles bears to the side adjacent to that angle, is the tangent of that angle.

Let CMP be any plane triangle, right-angled

P

In any plane triangle, the ratio of any two of the sides, is equal to the ratio

of the sines of the angles opposite to them.

Let ABC be a plane triangle; it is required to prove,

From C let fall CD perpendicular on AB.

Then, since CDB is a plane triangle right-angled at D, by last proposition,

CD = CB sin. B

........

Again, since CDA is a plane triangle right-angled at A,

In like manner, by dropping perpendiculars from B and A upon the sides AC, CB, we can prove,

In treating of plane triangles, it is convenient to designate the three angles by the capital letters A, B, C, and the sides opposite to these angles by the corresponding small letters, a, b, c. According to this notation, the last propo

Given the sines and cosines of two angles, to find the sine of their sum.

Let ABC be a plane triangle; from C let fall CD per

But, since ABC is a plane triangle, ++ C = 180°

(0 + 0).

sin. {180°(+0)}

= sin (+), because 180° (+) is the supplement of

Hence, sin. (+0) = sin. cos. + sin. cos. .......

This expression, from its great importance, is called the fundamental formula of Plane Trigonometry, and nearly the whole science may be derived from it.