use.

The following results are worthy of notice, as being of frequent

a = b. sin A

=b. cos C=c. tan A= c. cot C,
b=a.cosec A = a. sec C = c. sec A = c. cosec C,
c = a. cot A

= a. tan C = b. cos A=b. sin C.

63. The trigonometrical ratios remain unchanged so long as the angle is the same.

B

In QB take any points P, P' and draw PM, P'M' at right

angles to QE.

Then, since PM, P'M' are parallel,

the triangles QPM, QP'M' are similar.

i. e. the value of the sine of EQB is the same so long as the angle is the same, and this result holds good for the other ratios.

The figure represents the simplest case, where the given angle is less than a right angle, but the conclusion is true for all angles:

CHAPTER V.

ON THE CHANGES IN SIGN AND MAGNITUDE OF THE TRIGONOMETRICAL RATIOS OF AN ANGLE AS IT INCREASES FROM 0° TO 360°.

64. LET NS, WE bisect each other at right angles in the point, and let a line equal in length to QE be supposed to revolve from QE round the fixed point Q.

As the revolving line passes from the position QE to the positions QN, QW, QS, QE, the extremity traces out a circle ENWS.

If we take a succession of points in EN, as P, P, P, P, and from them let fall perpendiculars PM, PM, PM, PM on the line QE, and do the same in the other quadrants, it is clear that

In passing from E to N,

PM continually increases from zero to QN,
QM continually decreases from QE to zero.

In passing from N to W,

PM continually decreases from QN to zero,
QM continually increases from zero to QW.

In passing from W to S,

PM continually increases from zero to QS,
QM continually decreases from QW to zero.

In passing from S to E,

Again,

PM continually decreases from QS to zero,
QM continually increases from zero to QE.

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

QM is positive in the first and fourth quadrants,

negative in the second and third.

QP is always positive.

65. To trace the changes in sign and magnitude of the sine of an angle as the angle changes from 0° to 360o.

Let NQS, EQW be two diameters of a circle at right angles. Let a radius QP by revolving from QE trace out any angle PQE, and denote this angle by A.

From P draw PM at right angles to EQW.

As A changes from 0° to 90°, QP revolves from QE to QN;

As A changes from 270° to 360°, QP revolves from QS to QE; .. PM changes from QS to 0 and is negative,

66. To trace the changes in the sign and magnitude of the cosine of an angle as the angle changes from 0° to 360°.

Making the same construction as in Art. 65,

As A changes from 0° to 90°, QP revolves from QE to QN;

.. QM changes from QE to 0

QP

and is positive,

... cos A

As A changes from 180° to 270°, QP revolves from QW to QS;

.. QM changes from QW to 0 and is negative,