PLANE TRIGONOMETRY.

CHAPTER I.

GENERAL PRINCIPLES OF PLANE TRIGONOMETRY.

1. Trigonometry is the science which treats of augles and triangles.

2. Plane Trigonometry treats of plane triangles. [B. p. 36.]*

3. To solve a Triangle is to calculate certain of its sides and angles when the others are known.

It has been proved in Geometry that, when three of the six parts of a triangle are given, the triangle can be constructed, provided one at least of the given parts is a side. In these cases, then, the unknown parts of the triangle can be determined geometrically, and it may readily be inferred that they can also be determined algebraically.

But a great difficulty is met with on the very threshold of the attempt to apply the calculus to triangles. It arises from the circumstance, that two kinds of quantities are to

* References between brackets, preceded by the letter B., refer to the pages in the stereotype edition of Bowditch's Navigator.

Solution of all triangles reduced to that of right angles.

be introduced into the same formulas, sides, and angles. These quantities are not only of an entirely different species, but the law of their relative increase and decrease is so complicated, that they cannot be determined from each other by any of the common operations of Algebra.

4. To diminish the difficulty of solving triangles as much as possible, every method has been taken to compare triangles with each other, and the solution of all triangles has been reduced to that of a Limited Series of Right Triangles.

a. It is a well known proposition of Geometry, that, in all triangles, which are equiangular with respect to each other, the ratios of the homologous sides are also equal. [B. p. 12.] If, then, a series of dissimilar triangles were constructed containing every possible variety of angles : and, if the angles and the ratios of the sides were all known, we should find it easy to calculate every case of triangles. Suppose, for instance, that in the triangle ABC (fig. 1.), the sides of which we shall denote by the small letters a, b, c, respectively opposite to the angles A, B, C, there are given the two sides b and c and the included angle A, to find the side a and the angles B and C. We are to look through the series of calculated triangles, till we find one which has an angle equal to A, and the ratio of the including sides equal to that of b and c. As this triangle is similar to ABC, its angles and the ratio of its sides must also be those of the triangle ABC, which is therefore completely determined. For, to find the side a, we have only to multiply the ratio which we have found of b to a, that is,

a

a

the fraction by the side b or the ratio by the side c.

Solution of all triangles reduced to that of right triangles.

b. A series of calculated triangles is not, however, needed for any other than right triangles. For every oblique triangle is either the sum or the difference of two right triangles; and the sides and angles of the oblique triangle are the same with those of the right triangles, or may be obtained from them by addition or by subtraction. Thus the triangle ABC is the sum (fig. 2.) or the difference (fig. 3.) of the two right triangles ABP and BPC. In both figures the sides AB, BC, and the angle A belong at once to the oblique and the right triangles, and so does the angle BCA (fig. 2.) or its supplement (fig. 3.); while the angle ABC is the sum (fig. 2.), or, the difference (fig. 3.) of ABP and PBC; and the side AC is the sum (fig. 2.), or the difference (fig. 3.) of AP and PC.

c. But, as even a series of right triangles, which should contain every variety of angle, would be unlimited, it could never be constructed or calculated. Fortunately, such a series is not required; and it is sufficient for all practical purposes to calculate a series in which the successive angles differ only by a minute, or, at least, by a second. The other triangles can be obtained, when needed, by that simple principle of interpolation made use of to obtain the intermediate logarithms from those given in the tables.

1*

Sine, tangent, secant.

CHAPTER II.

SINES, TANGENTS, AND SECANTS.

5. Confining ourselves, for the present, to right triangles, we proceed to introduce some terms, for the purpose of giving simplicity and brevity to our language.

The Sine of an angle is the quotient obtained by dividing the leg opposite it in a right triangle by the hypothenuse.

Thus, if we denote (fig. 4.) the legs BC and AC by the letters a and b, and the hypothenuse AB by the letter h, we have,

6. The Tangent of an angle is the quotient obtained by dividing the leg opposite it in a right triangle, by the adjacent leg.

7. The Secant of an angle is the quotient obtained

by dividing the hypothenuse by the leg adjacent to the angle.

8. The Cosine, Cotangent, and Cosecant of an angle are respectively the sine, tangent, and secant of its complement.

9. Corollary. Since the two acute angles of a right triangle are complements of each other, the sine, tangent, and secant of the one must be the cosine, cotangent, and cosecant of the other.

10. Corollary. By inspecting the preceding equations (4), we perceive that the sine and cosecant of an angle are reciprocals of each other; as are also the cosine and secant, and also the tangent and cotangent.