PLANE TRIGONOMETRY

PART I

WITH THE USE OF LOGARITHMS.

BY THE REV. J. W. COLENSO, M. A

RECTOR OF FORNCETT ST. MARY, NORFOLK,

LATE PELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE.

Second Edition.

LONGMAN AND CO., LONDON.

MDCCCLI.

O APR 1961

LIBRARY

CAMBRIDGE:

PRINTED BY METCALFE AND PALMER.

FACTS AND FORMULE

To be committed to memory.

T= 3.14159=3} or is, nearly: circumference of circle=2πr, area=πr2.

=3

180

=57.29577 : w° (of arc)=radius; w° (of angle)=unit of circ. measure. The unit of circular measure is the angle which, in any circle, subtends at the centre an arc equal in length to the radius.

The sequence of signs in the four quadrants are

for sine and cosecant (++−−), for cosine and secant (+− − +),
for tangent and cotangent (+ − + −).

sinA=+sin(180°-A)=-sin (180°+A) = ~sin (-A)

cosA=-cos(180° — A) —cos (180°+A) = +cos(-A)

secA =

tanA=-tan(180°-A)=+tan(180°+4)=-tan(-A)

area S=3be sin^= √ {s(s—a)(s—b)(s—c)}: R=

(s—b) (s—c)

loga (a-1)-(a−1)2+(a−1)3-&c.

a2 = 1+ (loga) x + (logea)2 +&c, e=1+x+

L sin4=10+ log sin4; LcosecɅ=20-L sin; &c.

√2=1.4142...., √3=1.7320....

PLANE TRIGONOMETRY.

CHAPTER I.

ON THE MEASUREMENT OF LINES AND ANGLES.

1. TRIGONOMETRY, from Tpíywvov, a triangle, and μeтpéw, I measure, means properly the science which treats of the measurement of triangles, that is, of their sides, angles, areas, &c.; being called either plane or spherical trigonometry, according as the triangles are drawn upon a plane or on the surface of a sphere, in which latter case the sides will be circular arcs and the angles curvilineal.

But the word is now used in a much more extended sense, so as to include all manner of algebraical reasoning about lines and angles, whether parts of a triangle or not, and, particularly, when that reasoning is carried on by means of certain quantities, which are called the trigonometrical Ratios or Functions of an angle.

Of these Ratios we shall speak presently, so far as the subject of plane trigonometry is concerned. But we must first make some remarks upon the mode in which lines and angles are represented for the purposes of algebraical reasoning.

2. A line is represented algebraically by some letter, as a, which denotes the number of times it contains a certain line, taken as the unit of measurement.

Thus if the unit be a foot or an inch, then the line a would be one containing a feet or a inches.

B