PLANE TRIGONOMETRY.

CHAPTER I.

INTRODUCTION AND DEFINITIONS.

PLANE TRIGONOMETRY, as the name imports, was originally employed solely in determining, from certain data, the sides and angles of plane triangles. In modern analysis, however, its objects have been much extended, and the formulæ of this branch of Mathematics are extensively employed as instruments of calculation in almost every department of scientific investigation. From this circumstance, some writers wishing to change its designation to one which might more fully express its nature and applications, have proposed to term it the Arithmetic of Sines, others, the Calculus of Angular Functions, but the original appellation is still retained by the great majority of authors upon these subjects.

In treating of angular magnitude, we have hitherto confined ourselves to the consideration of angles less than two right angles; but in trigonometry it is frequently necessary to introduce angles which are greater than two, than three, or even than four right angles. We may take the following method of illustrating the generation of angular magnitude.

Let Aa be a fixed straight line, and let a line CP be supposed to revolve round the point C in Aa, and to assume in succession the different positions CP1, CP 2,

When CP coincides with CA, there is no angle contained between CP and CA, or the angle CAP is 0.

When CP begins to revolve round C, and comes into the position CP1, it forms with CA an angle P1CA less than a right angle.

P3

Pi

P6

P1

When CP has performed one-fourth part of an entire revolution, and has thus reached the position CP2, where CP, is perpendicular to CA, it forms with CA the angle P¿CA, which is a right angle.

As CP continues its revolutions, it will assume the position CP3, forming with CA the angle P3 CA, greater than one, and less than two right angles.

When CP coincides with Ca, it has performed one

half of an entire revolution, and forms with CA the

angle aCA, equal to two right angles.

CP having passed Ca, will assume the position CP 4, forming with CA the angle P4 CA, greater than two, and less than three, right angles.

The dotted space indicates the angle which we are considering.

When CP has performed three-fourths of an entire revolution, it assumes the position CP5, where CP, is perpendicular to Aa, forming with CA the angle P¿CA, equal to three right angles.

Passing beyond CP, the revolving line assumes the position CP6, forming with CA the angle P.CA, greater than three, and less than four, right angles.

P6

Finally, when the line CP has completed an entire revolution, it will return to its original position CA, having formed with CA an angle equal to four right angles.

If we suppose the revolution to recommence, it is manifest that CP may be conceived to form with CA angles greater than four, than five, or than any given number of right angles.

It is convenient in trigonometrical investigations, to draw two straight lines at right angles to each other,

and from their point of intersection to describe a circle, with any radius cutting these lines in any points A, B, a, b.

The circumference of the circle will thus be divided into four equal arcs AB, Ba, ab, bA, each of which, be

ing a fourth part of the whole circumference, is called

a quadrant, and subtends a right angle at the centre of the circle.

B

AB is called the first quadrant, Ba the second quadrant, ab the third quad

rant, and bA the fourth quadrant.

If each of these right angles be divided by straight lines CP1, CP2, whole circumference will be divided into a correspond ing number of equal parts, each of which is called a degree. The whole circumference will thus contain 360 degrees, and each quadrant will contain 90 degrees.

.... into 90 equal angles, the

The angles themselves, and the arcs which subtend them, are called degrees indifferently.

Angles are usually designated by the number of degrees which they contain; thus, a right angle is called an angle of 90 degrees; two right angles, an angle of 180 degrees, &c.

If each degree be divided into 60 equal parts, each of these smaller angles is called a minute.

If each minute be divided into 60 equal parts, each of these smaller angles is called a second.

Thus, four right angles, or the entire circumference of a circle, contains 360 degrees; 360 × 60, or 21,600 minutes; 360 × 60 × 60, or 1,296,000 seconds.

Degrees are expressed in writing by placing a small cypher immediately above the number to the right; thus 90°, 45°, 63°, signify 90 degrees, 45 degrees, 63 degrees, &c.

Minutes are expressed by placing one accent in the same manner above the number, and seconds by placing two accents: thus, 35′, 40′, &c. signify 35 minutes, 40 minutes, &c.; and 35", 40", signify 35 seconds, 40 seconds, &c.

Any lower subdivision of a degree is usually expressed in decimal parts of a second.

NN

Another division of the circle has been introduced by some modern authors, especially the French. They divide the whole circumference of the circle into four hundred equal parts or degrees, each degree into one hundred minutes, and each minute into one hundred seconds. This method possesses many practical advantages over the former; but the number of valuable tables calculated according to the former system, will in all probability prevent it from being generally adopted. It is called the decimal division of the circle, in contradistinction to the former which is called the sexagesimal division.

This being premised, we shall proceed to define the more important trigonometrical terms.

1. The complement of an angle is the defect of an angle from ninety degrees. Thus, if be any angle, the complement of ◊ is (90° — A).

2. The supplement of an angle is the defect of an angle from one hundred and eighty degrees. Thus, if be any angle, the supplement of ◊ is (180° — ◊). Draw two straight lines Aa, Bb, at right angles to each other, intersecting in the point C. With centre

C and any distance as radius, describe a circle, cutting the straight lines in the points A, B, a, b.

Draw the radius CP1, forming with CA any angle ACP1 = 0.

From P1 draw P1 M1 perpendicular on Aa,

1

From A draw AT, a tangent to the circle at A.

Produce CP to meet AT, in T1.

B

P., Ti

MA

3. Then the ratio of P, M1 to the radius of the circle, is called the sine of the angle P1CA.

4. The ratio of AT, to the radius of the circle is called the tangent of the angle P1CA.

Or,

ᎪᎢ,
CA

= tan.

5. The ratio of CT, to the radius of the circle is called the secant of the angle P1CA.

Cr,

ст,
CA

= sec. 8

6. The ratio of AM, to the radius of the circle, is called the versed sine of the angle PICA.

Or,

AM1
CA

= v. sin.

7. The sine of the complement of any given angle is called the cosine of that

angle.

Or,

sin. (90°) = cos.

and .. cos. (90°

@) = sin. 8. 8. The tangent of the complement of any given angle is called the cotangent

9. The secant of the complement of any given angle is called the cosecant of

10. The versed sine of the complement of any angle is called the co-versed sine of that angle.

-

Or, v. sin. (90° 0) = co-v. sin. and.. co-v. sin. (90° — 4) = v. sin. @ We shall now prove that the ratio of CM, to the radius of the circle, in the last figure, is the cosine of the angle PCA, that is, the sine of its complement.

Draw a circle A'B'a'b', equal to the circle ABab, and from C' the centre draw C'P' making with C'A' the angle P'C'A' equal to the angle PCB, i. e. to the complement of P, CA, or to (90° — A).

1

Then since CP, is equal to CP, and the angles at M, and M' right angles, and the angle CP, M, equal to the angle P'C'M', the two triangles P,CM1, P'CM', are equal in every respect, P1M1 = C'M', CM, = P'M'.

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

Thus, for example, let us take an angle P,CA situated in the second quadrant, that is, an angle greater than one right angle and less than two.

From P,let fall P,M, perpendicular on Aa, from a draw aT, a tangent to the circle at a meeting CP produced in T.; then as before,

B

T2 P

Again, let the angle in question be situated in the third quadrant, that is, let it be an angle greater than two and less than three right angles.

Making a construction analogous to that in the two former cases, we shall have

Lastly, let the angle be situated in the fourth quadrant; that is, let it be an angle greater than three and less than four right angles, then as before

We shall now proceed to establish some important general relations between the trigonometrical quantities which are immediately deducible from the above definitions, and from the principles of Geometry.

Resuming the figure of Def. (2):

Since CMP is a right-angled triangle and CP the hypotenuse,

PM2 + CM2 = CP2

BD

T

The triangles PMC, TAC, are equiangular and similar; hence

.....(2,

(3)

• The symbols sin. §, cos.2 0, tan. 3, &c., signify the square of sin. f, the square of cos. §, &c.; this is the common notation;—-another, more strictly in accordance with analogy, is sometimes employed to express the same thing, viz. (sin. f)a, (cos. f)a, (tan. e), &c.