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the knowledge of a trigonometrical line is equivalent to the knowledge of its arc or angle, and vice versa.

The trigonometrical lines are sometimes called trigonometrical functions of an arc or angle; the reason for which will be understood if we first explain the signification of the word function as employed in mathematics.

One quantity is said to be a function of another, when the former depends in any way upon the latter for its value.

It is said to be an increasing function when it increases as the quantity upon which it depends increases; and a decreasing function when it diminishes as the other increases.

The quantity upon which a function depends is called its variable, because this is supposed to change its value at pleasure, the function changing to correspond.

Now a trigonometrical line depends upon the magnitude of its arc for its length; it is therefore properly termed a function of the arc; and by way of distinction a trigonometrical function.

Of these trigonometrical lines, we now proceed to explain the nature and properties.

THE SINE.

14. The sine of an arc is a perpendicular let fall from one extremity of the arc upon the diameter drawn through the other extremity.

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ference have, it thus appears, the same sine. Two such arcs are

called supplements of each other. A semi-circle contains 180°. The supplement of an arc is therefore what is left after taking the arc from 180°. Thus 80° is the supplement of 100°. 70° is the supplement of 110°.

Two arcs 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 arc 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 dimi

M

nishes, i. e., becomes a decreasing function of the arc till the arc reaches 180°, when the sine is 0 again. Beyond this value of the arc the sine again increases till the arc reaches 270°,

The word proposition is here used in the enlarged sense of any thing propounded as true.

or three quadrants, when the sine is again equal in length to R.

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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 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 will be the arc in the first quadrant, having the same sine as the given arc,

For example, let the

given arc be 200°. This is nearest 180°, and differs 20°. The sine of 20° is equal in length to the sine of 200°. Or M P, which is the sine of A B M, is also the sine of B M. Again, let the given arc be 300°. This is nearest 360, and differs 60. The

BP

M

sine of 60° is equal in length to the sine of 300,

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 perpendicular 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 algebra, 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 student is referred to the well known problem of the Couriers, Art. 108. Davies' Bourdon.

THE TANGENT.

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

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arc is 0, the tangent will evidently be 0. As the arc increases the tangent 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.

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