Notice that when we have such a formula as

sin (A+B) + sin(A - B) = 2sinA cosB,

we may use it either to replace the sum of two sines by the product of the sine and cosine; or, by reading it from right to left, to replace the product of the sine and cosine by the sum of two sines. For example, we may either write

sin100+ sin60 for 2sin80 cos20;

or, sin80 cos20 for (sin100+ sin60).

Similarly, we should be ready to detect the formula under such sin100=2sin80 cose - sin60,

a form as

(6) We may easily remember the following formulæ by thinking of a very simple principle :—

sin (A+B) sin(A - B) = sin2A - sin2B=cos2B - cos2A. cos (A+B) cos(A-B) = cos2A - sin2B = cos2B - sin2A. Think of the expressions for sin (A+B) for the first formula, an and observe (a) that each product in the above formula is expressed by the difference of the squares of two functions of A and B ; (b) that the first function, sinA, or cosB, is taken out of sinA cosB, the first term of the expressions for sin (A + B); and the second function, sin B, or cosa, is taken out of cosA sinB, the second term of the expressions for sin (A+B).

Similarly think of the expressions for cos (A+B) for the second formula, and proceed on the same simple analogy.

(7) The student should observe that any formula which has been obtained for functions of A+B and A - B in terms of A and B, may be changed into a corresponding formula for functions of A and B in terms of (A+B) and (A-B), by simply writing A for A+B, B for A-B, (A+B) for A, and (A-B) for B. Let the student test this by substitution in all the formulæ implied in (5).

(8) He will also observe that he may express a product of two sines, or of two cosines, or of a sine and a cosine, either by means of a sum or difference, or by means of the difference of two squares; and conversely.

The converse is very important; for it will enable us to adapt certain given quantities to logarithms by means of factors.

(9) By means of some of the above derived formula we may obtain the values of the trigonometrical ratios of those angles which are the halves of 30°, 45°, 60°, &c.; and then further of the halves of these angles. We may also obtain the values of the ratios of 9°, 18°, 36°, 54°, 72°, 75°, 81°, and even of 3o, 6o, 12°, &

25. Before passing on to the next section of the subject the student may consider the two following propositions, although they are not essentially connected with the chief requirements of the Pass Examination :

If be the circular measure of an angle less than a right angle, O is greater than sino and less than tano.

sino

The limit of 0 when 0 is indefinitely diminished is unity.

RELATIONS BETWEEN THE SIDES AND ANGLES OF A

TRIANGLE.

Before we proceed to consider the “several cases of plane triangles" it is necessary to investigate the relations between the sides of a triangle and the trigonometrical ratios of the angles of the triangle. It is these relations and the formulæ expressing them which are applied to the solution of triangles.

26. To show that in any triangle the sides are proportional to the sines of the opposite angles.

27. To express the cosine of an angle of a triangle in terms of the sides.

28. In every triangle each side is equal to the sum of the product of each of the others into the cosine of the angle which it makes with the opposite side.

29. To express the sine, cosine, and tangent of half an angle of a triangle in terms of the sides.

31. To deduce from Article 29 or from Article 27 the value of sinA in terms of the sides of a triangle.

32. Having given any one of the three formula implied in Articles 26-28 to deduce the other two formulæ.

THE SOLUTION OF TRIANGLES.

The "several cases are given with such fulness of detail in every satisfactory work on Trigonometry that we may safely leave the student with his text-book, simply reminding him that he cannot study this branch of his subject too carefully and thoroughly, and that he must pay particular attention to the discussion of the Ambiguous Case which always occurs where we have given two sides and the angle opposite to one of them (that is, an angle not included by the two given sides), and the side opposite the given angle is less than the other given side.

The three formulæ which it is essential to remember are :

Sin A sinB

b

=

sinC

C

b2+c2- a2

which we may call the SINE-FORMULA.

which we may call the COSINE-FORMULA. , 2bc

(1)

α

(2) CosA=

(3) Tan

A-B a-b
2
a+b
FORMULA.

The last refers specially to the important case when we have given two sides and the included angle, and in connection with this case the student should investigate the nature and use of what is called a Subsidiary Angle.

The candidate cannot have too much practice with examples on the solution of triangles. Such examples will best familiarise him with the use of the formula and the application of Logarithms-an essential and most instructive part of the subject.

THE DETERMINATION OF THE AREAS OF TRIANGLES.

33. The following are the three expressions for the area (▲) of a triangle which the student should know how to find :

(1) A ac sinBab sinC=bc sinA.

Thus we can determine the area when we can find the length of any two sides and the sine of the angle between them.

(2) A =√s(sa) (s—b) (s−c) = {\/2a2b2+2a2c2+2b2c2—aa − ba—ca. Hence we can determine the area when we know the lengths of all the sides.

Hence we can also determine the area when any side and any two angles are given; for when two angles are known the third angle may be found. In the above formulæ sinB=sin(A+C), sinC=

sin(A+B), and sinA=sin(B+C).

The following are examples for practice :—

Ex. 1. In any triangle, right angled at C, obtain the following expressions for the area: ▲=1(a+b+c)(a+b−c); ^ =‡c2sin2A.

a2-b2 sin A sinB

Ex. 2. In any triangle show that ▲ = 2 sin(A–B);

A =

2abc a+b+c

cos Acos Bcos C.

Ex. 3. Show that the area of a triangle=4(a2sin2B + b2sin2A). Ex. 4. In a right-angled triangle the area=s(s—c), when C is the right angle.

a2 b2

Ex. 5. In any triangle, ▲ = sin^ sinsin (n+in+ sinc).

B 2 2

sinB +

Ex. 6. Prove the following expressions for the area of a triangle:

(a2bc cosB cosC+b2ac cosa cosC+c2ab cosA cosB).

THE DETERMINATION OF HEIGHTS AND DISTANCES.

34. To find the height and distance of an inaccessible object on a horizontal plane.

35. To find the distance between two visible but inaccessible objects and the distance of either from the place of observation.

36. To find the distance of a ship from the shore.

37. The lengths of the lines which join three points A, B, C are known; at any point P in the same plane as A, B, C the angles AP C, BP C are observed; it is required to find the distance of P from each of the points A, B, C.

The above are the problems, in determining heights and distances, which are most frequently required, and which furnish illustrations of the practical way of applying the formulæ for the solution of triangles. They should all be mastered, except perhaps the last, which is rather difficult, and probably beyond the requirements of the Pass candidate. The more exercises, in this part of the science, the student can perform and the better. They will show him the variety of applications of which each proposition is capable, and will satisfactorily test his comprehension of all the previous parts of the subject.

TEXT-BOOKS IN PLANE TRIGONOMETRY.

On the ground of clearness, simplicity, and adaptability to the needs of every beginner we place first Hamblin Smith's Elementary Trigonometry (4s. 6d., Key 7s. 6d., Rivington). It is not only rudimentary enough for the beginner, but comprehensive enough to meet all the ordinary wants of the candidate for the First B.A. It is well classified and graduated; it contains numerous explanations and worked examples; the exercises are very suitable and abundant; they are placed where they will be most readily understood and where they will best illustrate what has just preceded them. There is scarcely any part of the book which the candidate could safely omit; there seems

to be enough to cover all the real requirements of the examination; and for these and other reasons we cordially recommend the use of the book especially to those commencing the study of Trigonometry for the first time.

On similar grounds we can also commend Beasley's Elementary Treatise on Plane Trigonometry (3s. 6d., Macmillan). It is scarcely so full in its details, in its classification of formulæ, and its illustrative or worked examples, as the former work; but its method is throughout good, its reasoning concise and easy to understand, its very numerous exercises are well classified according to their relative difficulty, and they are supplemented by miscellaneous examples and Examination-Papers. The reader of it will observe that the consideration of Circular Measure is postponed till near the end of the book, and that the subject of Heights and Distances, instead of being distinctly treated in a separate chapter, is illustrated and applied throughout the book.

In Hann's Elements of Plane Trigonometry (1s., Lockwood and Co.) we have a little, yet useful and practical, treatise containing all the essential parts of the subject in outline, and illustrating these parts by very many worked examples which will be of considerable service to the self-taught student. Its earlier portions are somewhat wanting in explanatory remarks, but this defect is more than covered by the illustrations given in the examples which are worked out at full length and which plainly indicate the many applications of which the formulæ are capable. It is a good work to take in conjunction with a better classified and more comprehensive treatise. It does not, like the other works referred to, include the subject of Logarithms.

In both Colenso's Plane Trigonometry, Part I. (3s. 6d., Key 3s. 6d., Longmans), and Todhunter's larger work, Plane Trigonometry for the Use of Colleges and Schools (5s., Key 10s. 6d., Macmillan), there is an ample treatment of everything required in the Pass Examination. These are text-books in very general use, and either is sufficient in itself, although, of course, Todhunter's work is by far the more comprehensive and exhaustive. It is perhaps the more difficult of the two for a mere beginner; but this obstacle may be overcome by a previous perusal of the same author's Trigonometry for Beginners (2s. 6d., Key 8s. 6d., Macmillan), which is designed to serve as an introduction to the larger treatise, but is scarcely advanced enough to be taken by itself. In the larger work the various formulæ and propositions are shown to be universally true-a point the more elementary treatises do not pretend fully to discuss. Colenso's work, we may add, is remarkably helpful, suggestive, and compact. Part II. (2s. 6d., Key 5s., Longmans) is beyond the requirements of the Pass Examination. It treats on the Summation of Series, the Trigonometrical Solution of Equations, and includes a large collection of Miscellaneous Problems.