CHAPTER VIII

TRIGONOMETRY

TRIGONOMETRY treats of the measurement by calculation of the different parts of a triangle. It is a branch of mathematics which is essential to Surveying.

Every triangle has six distinct parts, three angles and three sides. If we know any three parts (except the three angles) we can ascertain by calculation the remaining three parts. We could, of course, ascertain the three remaining parts by plotting' and scaling also, but this method is not sufficiently accurate, and so we resort to trigonometry, which enables us to calculate without plotting

any of the parts which we could have ascer tained by plotting and scaling.

The difficulty is to obtain a common basis of measurement because angles are expressed in degrees, minutes, and seconds (a right angle being 90°) and sides according to the ordinary standard of linear measurement such as chains, yards, feet, &c.

It will be perfectly obvious that in a sum involving the multiplying together, &c., of such terms as degrees and minutes by chains, or yards, we should not advance much towards a solution.

It is necessary to obtain some means whereby we can express angles and sides in the same terms. Indirectly angles may be expressed in terms of linear dimensions by means of ratios. In the case of any angle, if we drop a perpendicular from one of the two lines containing it to the other line, any two sides of the triangle thus formed will always bear to one another a fixed proportion or ratio. By means of these ratios we can represent an angle in a calculation which contains linear dimensions.

There are six ratios which can be formed out of the sides of a right-angled triangle. They are called Functions and are as follows:

It should be very carefully noted that in right-angled triangles the side opposite the right angle is called the Hypotenuse and the remaining sides the base and perpendicular. Now in trigonometry the side opposite the angle which is being used in the solution of the triangle is called the Perpendicular.

Thus in Fig. 129 the side AC is the perpendicular with respect to the angle B, but with respect to the angle A, BC is the perpendicular and

It is important to bear this in mind as the beginner very often loses sight of it. The functions must, of course, be learnt by heart, but the following mnemonic will be of great assistance :-Peter Has Been Here Playing Billiards.

=

=

Taken in pairs and using the first letter of each word we get Peter Has, Perpendicular over Hypotenuse sine: Been Here, Base over Hypotenuse cosine: Playing Billiards, Perpendicular over Base tangent. It will be seen that the cotangent, secant, and cosecant are the reciprocals of the tangent, cosine, and sine respectively.

Before proceeding further let us first take note of the following facts:

(a) All three angles of a triangle together equal 180°. (b) In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. (c) The complement of an angle is that which must be added to it in order to make a right angle.

(d) The supplement of an angle is that which must be added to it in order to make two right angles.

In a right-angled triangle, therefore, if one of the acute angles be known, the other, its complement, must be the difference between it and 90°; and if any two angles of a triangle be known, the third must be 180 minus their sum.

The value of any function can be found if the number of degrees be known or the number of degrees can be ascertained if the function be known. For example, an angle may be described as such that its tangent is. Now tangent means perpendicular divided by base, consequently the angle is such that the base is twice the length of the perpendicular. By plotting a triangle with a base twice as long as the perpendicular the angle is obtained whose tangent is. With a protractor the angle would be found to contain about 26° 34'. But by means of Chambers' 'Mathematical Tables' we are able to find out the numerical values of the functions of angles.

How to use the Tables of Functions of Angles.-Sines and Cosines. On page 263 will be found columns headed Natural

Sines, Cosines, Chords, &c.' At the top of the page is 0° and at the bottom 89°. On the extreme left is a column of minutes reading from 0 to 60 downwards, and on the extreme right is a similar column reading upwards. It will be also noticed that the column headed 'Sine' is called 'Cosine' at the bottom and vice versa: and the number of degrees at the top is just one less than the complement of the number of degrees at the bottom of the page.

What is the sine of 2° 16′?-Look on the page headed 2° and follow down the column of minutes until you come to 16the figures in the sine column against this number are 0395505. All the figures given are decimals, so a decimal point must be understood where it is not shown.

What is the cosine?-Look down the column headed 'cosine,' using the same minute column, and the figures found are 9992176.

... cos 2° 16′ = 0.9992176

Note very carefully that when the number of degrees appears at the top of the page the left-hand minute column must be used, and when the number of degrees appears at the bottom of the page the righthand minute column must be used, always. This is very important.

What is the sine of 87° 44′ ?—This time the number of degrees appears at the bottom of the page (as do all the numbers from 45° to 90°) so that we must use the right-hand minute column.

What is the cosine? Look up the column marked cosine and using the same minute column-that is, the right-hand one-we find that

cos 87° 44' = 0.0395505

= the cosine of

It will be seen from this that the sine of an angle its complement, for 87° 44′ is the complement of 2° 16'. By printing the work so as to read from the top and the bottom, the publisher thereby saves space.

Tangents, Cotangents, Secants, and Cosecants.-These are more simple to read. The tables follow immediately after those just considered-namely, sines and cosines.

In ordinary work it is not usually necessary to read an angle more precisely than to the nearest minute, but if it is required to read to seconds, the method of finding out the logarithmic functions is shown on page xxii in the Tables.

Solution of Right-Angled Triangles

RULE I.- If the length of a side be required, place this side over one of which the length is known, and state what function it makes. Example.-Solve the triangle ABC (Fig. 130), when the angle C = 90°, A = 32° 32', and AC 40 feet.

Find the length of BC first. Therefore, place BC over AC whose length is known and state the function it makes. BC is the perpendicular (A being the angle in question, or that which we are using in the solution) and AC is the base.

Perpendicular Base = tangent

40 ft.

FIG. 130.

Now look in the tables for the numerical value of tan 32° 32'.

The angle B must, of course, be 90°

= 57° 28'.

The three unknown parts have thus been found and the triangle solved.

RULE II.- If an acute angle be required and the other be not known, form a vulgar fraction with the two

given sides and state, as an equation, what

function it represents.

Solve the triangle ABC (Fig. 131)

B

.. 1.3333333 = tan A

Look in the tables of natural tangents for these figures, and note what angle it represents. The figures are given in numerical order and the nearest to 1.3333333 is 1-3334900 which is against the angle 53° 8'.

... the angle A = 53° 8'