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ANALYTICAL PLANE TRIGONOMETRY.

CHAPTER I.

PLANE TRIGONOMETRY is the science which treats of the relations of the sides and angles of plane triangles.

In every triangle there are six parts: three sides and three angles; which have such relations to each other that the value of one depends on the value of the others; and if a sufficient number of these are known the others may thereby be determined.

The sides of triangles consist of absolute magnitude, but the angles are only the relations of those sides to each other in position or direction, without regard to their magnitudes.

Angles have no absolute measure in terms of the sides; but are, nevertheless, susceptible of measure; for if two lines meet each other the space included between them within a given distance from their point of contact is proportional to their mutual inclination, and hence (Prop. XVIII. Cor. B. III. El. Geom.) the arc of the circumference of a circle intercepted by two lines drawn from its centre, may be regarded as the measure of the angle or inclination of those lines, and therefore the arc of the circumference may be regarded as the measure of angular magnitude.

For this purpose the circumference of the circle is supposed to be divided into 360 equal parts, called degrees, and each of those degrees is divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds; and so on, to thirds, fourths, &c.

These divisions are designated by the following characters, • &c. Thus the expression 30° 20′ 12" 22""', represents an arc or an angle of 30 degrees 20 minutes 12 seconds 22 thirds.

The circumference of any circle may in this manner be applied as the measure of angles, without regard to its magnitude or the length of its radius; hence a degree is not a mag

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nitude of any definite length, but is a certain portion of the whole circumference of any circle, for it is evident that the 360th part of the circumference of a large circle is greater than the same part of a smaller one, but the number of degrees in the small circumference is the same as in the large one. The fourth part of the circumference of a circle is called a quadrant and contains 90 degrees: hence 90 degrees is the measure of the right angle.

Thus, if we draw two straight lines AD, BE, so as to cross each other at right angles, and from their point of intersection, C, we discribe a circle with any radius so as to cut those lines in any Apoints, as a, b, d, e, the circumference of the circle will thus be divided into four equal arcs, ab, bd, de, ea, each of which measures or subtends a right angle at the centre C, of the circle.

Pa

P

B

B

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If a line CP be made to revolve round a fixed point C as the centre of a circle, and so as to pass successively through every point of the circumference, commencing in the point a, then, while it is in the position Ca, or while it coincides with the line Ca, those two lines form but one, and intercept no arc on the circumference of the circle, and hence form no angle with each other; but when the line CaP comes into the position CP, it forms with AC an acute angle at C, which is measured by the arc aP, and when it comes into the position CbP, it then forms a right Aangle ACP with the line AC, which angle is measured by the quadrant ab. Now let it come into the position CP,, and the angle which it forms with CA, will be measured by the arc aP,, which is greater than a right angle, and hence is an obtuse angle.

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E

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Let it now come into the position CdP; it then coincides with the right line Cd, which is a portion of the line AC produced, since the line CP, in this position, coincides with the line AD, it can be said to form with it no angle; yet the space passed over by the line CP, from the position CaP, is equal to two quadrants, or two right angles equal to 180 degrees, and for trigonometrical investigation the lines CbP and CA are said to subtend the angle measured by the arc abd.

After passing the point d, and coming into the position CP,, it forms with AC, and on the upper side of it the angle P,CA

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measured by the arc aeP,, but having passed over the arc abdP,, is said to contain, with the line CA, the angle ACP, on the upper side of those lines measured by the arc abd P, greater than two right angles. When it comes in the position CeP it is said to subtend, with the line AC from the same side of it, the angle measured by the arc abde, or three quadrants, equal to three right angles.

When in the position CP,, it is said to contain with CA, and on the same side of it, an angle greater than three right angles. Finally, when the line CP has completed an entire revolution, having returned to its original position, CA, it will have formed an angle with it equal to four right angles.

If the line CP continues to revolve, it is manifest that the angle will increase, and may with this view form with CA, angles greater than four, than five, or than any given number of right angles.

ab is called the first quadrant of the circle, bd the second, de the third, and ea the fourth quadrant.

It must be borne in mind, that the line CP cannot, geome. trically, be said to contain with another line, AC, an angle greater, nor quite equal to, two right angles, but in view of its supposed motion round one of its extremities, C, as a centre, it is said to contain, with the line AC, all the angular space through which it has passed in its revolution.

Thus, let the line CP have performed one complete revolution, from the position CaP to the same position again, the angle which it forms with the line CA, though absolutely nothing, is in view of its supposed motion measured by the quadrants ab+bd+de+ea, each of which quadrants are readily recognized as being contained by their several lines of division, when by removing those lines of division of the circumference, those several angles are all converted into one containing the whole circumference; hence, in view of this motion or relation of the two lines, they are said to contain an angle measured by the whole circumference.

DEFINITIONS AND ILLUSTRATIONS.

The following symbols are sometimes used.

1. The complement of an arc or of an angle, is what remains after taking that arc or that angle from 90 degrees. Thus, if be any arc or angle, the complement is 90°

2. Supplement of an arc or an angle, is what remains after taking that arc or angle from two right angles, or 180 degrees. Thus, if be any arc or angle, the supplement of is 180°-0.

If AP be any arc, and ACP be any angle, measured by that arc, then the complement of the angle is the angle PCB measured by the arc PB, and the supplement of the angle A is the angle PCD measured by the arc PBD, and if BCP is any angle measured by the

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arc PB, then PCA is the complement of 4, and if PCD is any angle, then will PCA be its supplement.

3. To represent the ratios of the sides and angles of triangles, right lines are drawn in and about a circle called sines, tangents, secants, &c.

Draw two right lines AD,BE cutting each other at right angles in the point C, with the centre C and any distance as radius, describe a circle cutting the lines in the points A, B, D, E.

Draw the radius CP forming with CA any A angle ACP 8. From P draw PS perpendicular on CA. From A draw AT a tangent to the circumference at A. Produce CP to meet AT in T.

T

P

B

E

C

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4. Then the ratio of PS to the radius CA of the circle, is called the sine of the angle PCA.

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5. The ratio of AT to the radius CA of the circle, is called the tangent of the angle TCA or PCA.

Hence,

AT

=tan. 8.

CA

6. And the ratio of CT to the radius is the secant of the angle PCA.

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7. The ratio of AS to the radius of the circle, is called the versed sine of the angle PCA.

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8. The sine of the complement of an angle, is called the sine complement, or cosine of that angle.

Thus, sin. (90°-4)=cos. 4, hence cos. (90°-4)=sin. 4. 9. The tangent of the complement of any given angle, is called the cotangent of that angle.

Thus, tan. (90°-4)=cot. &, hence cot. (90°-4)=tan.

10. The secant of the complement of any given angle, is called the cosecant of that angle.

Or, sec. (90°-4)=cosec. 4, hence cosec. (90°—◊)=sec. 4. 11. The versed sine of the complement of any angle, is called the co-versed sine of that angle.

Or.

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v. sin. (90°-8)=co-v. sin. 8,

and hence co-v. sin. (90-0)=v. sin. 0.

In order to show that the ratio of CS to the radius of the circle in the last figure, is the cosine of the angle PCA; that is, the sine of its complement,

Or that

CS
=cos. 0,
CA

Draw a circle A'B'D'E' equal to the circle ABDE, and from C' the centre, draw C'P', making with C'A' the angle P'C'A' equal to A the angle PCB; that is, to the compliment of PCA, or to (90°——◊).

Then, since CP is equal to C'P', and the angles at S and S' are right angles, the angle CPS equal to the angle P'C'S' the two trian- A gles PCS, P'C'S' are equal in every respect; PS=C'S', CS=P'S'.

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We have hitherto considered an angle PCA less than a right angle, but the same definitions are applied, whatever may be the magnitude of the angle.

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