Two arcs, then, which are supplements of each other, have the same sine, or, as it is sometimes expressed, the sine of an arc is equal to the sine of its supplement.

If a represent an arc of any number of degrees, the notation employed to express the sine of that are is sin a. The proposition* above, stated algebraically, will stand thus, sin a sin (180°-a.)

=

The sine of an arc is also the sine of the angle measured by that arc.

16. When the arc is very small, it is plain that its sine will be very small also, and that when the arc is 0, the sine will be 0.

As the arc increases the sine increases till the arc is 90°, which, being a quarter of the circumference, is called a quadrant, the sine of which is R. (R signifying radius; which line this letter, whenever employed hereafter, will be understood to represent.)

As the arc increases beyond 90°, the sine diminishes, i. e. becomes a decreasing function of the arc, till the

M

arc reaches 180°, when the sine is 0 again. Beyond this value of the arc the sine again increases till the arc reaches 270°, or three quadrants, when the sine is again equal in length to R.

From 270° to 360° the sine decreases, till at the latter value it is a third time 0. Beyond 360° we pursue the same round again, and no new variations are developed.

17. The least value of the sine is 0. It has this value at 0°, at 180°, and at 360°.

The greatest value of the sine is R. It has this value at 90° and at 270°. It has all possible values between 0 and R, but it has no different

The word proposition is here used in the enlarged sense of anything propounded

as true.

values, as the arc increases to two, three, and four quadrants, from those which it had in the first. So that when the sine of an arc greater than 90° is required, an arc, having an equal sine, may be found in the first quadrant. To find this arc we have the following rule, the correctness of which the annexed diagram will show. Observe how many degrees distant the termination of the given arc is from 180° or 360°, according to which of these two is nearest, and that number of degrees and fractions of a degree, will be the arc in the first quadrant, having the same sine as the given arc.

For example, let the given arc be 2000. This is nearest 180°, and differs 20°. The sine of 200 is equal in length to the sine of 200°. Or м P, which is the sine of A B м, is also the sine of в M.

Again, let the given arc be 300°. This is nearest 360°, and differs 60°. The sine of 60° is equal in length to the sine of 300°.

B

P

A

M

If the given arc exceeds 360°, subtract 360, and then apply the rule just given. If the arc contains a number of circumferences, divide by 360, and apply the rule to the remainder.

18. It is customary, for the purpose of being able to bring the trigonometrical lines as they appear in the figure, the more readily before the mind when the figure is not present, to begin all arcs at the same point; and the point commonly chosen is the extreme right of the circumference, determined by the intersection of the horizontal diameter of the circle with the circumference. This is the point a, in the last figure. An arc of 90° will then reach to the top of the circle, or the upper extremity of a vertical diameter. An arc of 180° will terminate at the left of the circle, or of the horizontal diameter. An arc of 270°, at the lowest point of the circle, or lower extremity of the vertical diameter. An arc of 360°, at the right of the circle, or point of beginning.

One advantage of this plan will readily appear. Since the arc always commences at the same point, namely, the right of the circle, the horizontal diameter will be the diameter which passes through one extremity of the arc, and wherever the arc may terminate, the perpendicular from the other extremity of it, which is the definition of the sine, will be a per

pendicular to the horizontal diameter; so that the sines of all arcs, in a diagram so constructed, will be perpendiculars to the horizontal diameter. The sines of arcs between 0° and 180° will be drawn downwards; and those of arcs between 180° and 360° will be drawn upwards.

According to the general principle of analysis, that quantities estimated in a contrary sense are distinguished by contrary signs, if the sines of arcs between 0 and 180° be considered as positive, those of arcs between 180° and 360° must be regarded as negative.*

THE TANGENT.

19. The tangent of an arc is a perpendicular drawn to the radius at one extremity of the arc, and terminated by the radius produced, which passes through the other extremity.

increases, and very rapidly as the arc approaches 90°. In order to trace the tangent through its various changes, we shall suppose the arc to commence at the point on the extreme right of the circle, and the degrees to be counted upwards, towards the left, as in a former case—the tangent of every arc will then be drawn at the extremity of the horizontal radius on the right of the centre, and be terminated by the radius produced, passing through the other extremity of the arc, which extremity will vary its position as the arc varies its magnitude.

See Algebra, page 182.

The tangent of an arc, terminating in the second quadrant, will be cut off below the origin* of the arc. Thus AT is the tangent of A M; and according to the principle adopted when treating of the sine, this tangent, being in the opposite direction to that of the tangent of an arc in the first quadrant, is negative.

When the arc is 180°, the negative tangent, which became shorter and shorter as the second extremity of the arc approached this point, again reduces to 0. Beyond 180°, or in the third quadrant, the tangent is cut off above the origin again. Thus A T in the annexed diagram, is the tangent of the arc A B M. The tangent of an arc in the third quadrant is, therefore, positive. When the arc is 270° or 3 quadrants, the tangent becomes parallel to the radius which produced

M

B

M

A term applied to the point A, where the arc commences.

first quadrant; and the same rule applies to finding the length of the tangent belonging to any given arc, from that of an arc in the first quadrant, as was given for the sine.

The tangent changes its sign in every quadrant, that is four times in going round the circle. It is positive in the first and third, two diagonal quadrants, and negative in the second and fourth, the other two diagonal quadrants.

The tangent is at the top and bottom of the circle, and 0 on the right and left.

THE SECANT.

21. The secant of an arc is a line drawn from the centre of the circle to the extremity of the tangent.

In the preceding diagrams, cr is the secant of the arc AM. It is also the secant of the angle measured by the arc.

As the arc with its tangent diminishes, the secant diminishes; and when the arc and tangent are 0, the secant is equal to R. The secant can never be less than radius, because the tangent cannot pass

within the circum

ference, and consequently the line from the centre to the extremity of the tangent, must extend at least to the circumference. When the arc is 90° the secant is ∞. When the arc is 180° the secant is R. And when the arc is 270° or three quadrants, the secant is again ∞. appear from an inspection of the last diagrams.

All which will

Zero may have

* The infinity here has the doubtful or double sign. always the double sign±0. Infinity only when it is the transition from a + to a