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ELEMENTARY MENSURATION.

[CHAP.

NOTE. The above definition refers only to a right circular cone. Cones may exist whose axes are not perpendicular to their bases, and whose bases are not circular. Such cones, however, are beyond the scope of the present text-book.

101. To find the surface and volume of a cone.

In the adjoining figure let the height AB=h units of length, the radius of the base BC=r, and the slant height AC=1.

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The whole surface of cone curved surface + area of base

=Trl+πr2

=πr (1+r).

Again, it may be shewn that the volume of a cone is one-third the volume of the cylinder on the same base and of the same height;

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Now Ready. Second Edition. Globe 8vo. Price 2s. 6d.

THE

TRIGONOMETRY

OF

ONE ANGLE

BY

REV. J. B. LOCK, M.A.

FELLOW AND BURSAR OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE
FORMERLY MASTER AT ETON

London

MACMILLAN AND CO., LIMITED

NEW YORK: THE MACMILLAN COMPANY

All rights reserved

PREFACE TO THE SECOND EDITION

THIS little book has been entirely rewritten for this Second Edition. It now contains (without the Appendix) only that portion of the Trigonometry of One Angle which is required in the Cambridge Previous Examination. It is hoped that it may be found useful as a very simple introduction to the study of Trigonometry.

SCHOOLMASTER.-"The selection has been judiciously made, and in parts re-written, especially when the author felt that by so doing he could improve it. We can well recommend the book."

SCHOOL GUARDIAN.—"The book is, as we should have expected, skilfully compiled, and likely to be of considerable service to private students."

OXFORD MAGAZINE.-"It should be of especial assistance to students commencing mechanics . . . It may with confidence be said that no boy is likely to make any effective progress in such a subject, who is unable to master a work in which the explanations and general arrangement are so clear as in that before us.

CAMBRIDGE REVIEW.-"It is very well printed and the figures are clear."

NATIONAL OBSERVER.-"Teachers with dull pupils will find his work most useful."

INDIAN ENGINEERING.-"It is not necessary to discuss this book at any very great length. It is intended for those students who require a knowledge of the properties of sines and cosines for use in the study of elementary mechanics. The properties of the Trigonometrical Ratios are so simple, and a knowledge of them is so easily acquired, that this little work will supply a suitable text-book for those students who have no leisure to study the more difficult properties of compound angles. But although the work deals with the most elementary parts of the subject, the explanations are throughout eminently scientific, and we are glad to see that the author has explained the application of the signs plus and minus from the consideration of angles and lines as displacement of direction and of distance respectively. Great care seems to have been taken to insure accuracy; and, from beginning to end, a teacher would find it hard to add much in the way of supplementary explanation. Mr. Lock's books are remarkable for the clearness and precision of the statements contained in them; the present work fully sustains the deserved reputation of its author."

DUBLIN EVENING MAIL.-"The propositions are properly arranged and clearly stated. The examples are numerous and judiciously graduated. The diagrams and print are remarkably clear and indeed beautiful."

SOLUTION OF RIGHT-ANGLED TRIANGLES.

27

Example 1. In a right-angled triangle, in which C is the right angle, A is 30° and BC is 10 ft. ; solve the triangle.

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.. B=60°, AC 17 320... ft., AB 20 ft.

Example 2. In a right-angled triangle, in which C is the right angle, A is 30° 14′ 30′′ and AC is 134 2 ft.; solve the triangle, it being given that tan 30° 14′ 30′′ = 5831.

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c=√(a2+b2)=√ √{(78·25)2 + (134·2)2} = √(6123·0625 +18009·64) = 155·3... Hence B = 59° 45′ 30′′; BC=78.25... ft.; AB=155.3 ft.

* In practice a book of Mathematical Tables is used, from which (as from a Dictionary) the numerical value of the ratio of any given angle can be found, or from which the size of the angle having a given ratio can be found. In Examination Papers if a book of Tables is not supplied, the necessary quotations are given from the Tables (as in the Example above).

THE SUPPLEMENT OF AN ANGLE.

59

67. To trace the changes in sin A as A increases from 0° to 180°.

Let a revolving line OP, of constant length (r), turning about 0, make with OR the angle ROP (A), so that the angle ROP increases continu

ously from 0° to 180°.

Draw PM perpendicular to

OR or RO produced.

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Then as ROP increases from 0° to 90°, MP increases continuously from 0 to r; .. sin A increases continuously from 0 to 1 and as ROP increases from 90° to 180°,

;

MP decreases continuously from r to 0;

.. sin A decreases continuously from 1 to 0.

68. To trace the changes in the sign and magnitude of cos A as A changes from 0° to 180°.

Let a revolving line OP, of constant length (^), turning about 0, make with OR the angle ROP (A), so that the angle ROP increases continuously from 0° to 180°.

Draw PM perpendicular to OR or RO produced.

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Then as the angle ROP increases from 0° to 90°, OM decreases continuously from r to 0;

..cos A decreases continuously from 1 to 0;

and as ROP increases from 90° to 180°, OM' is drawn in the opposite direction, and its measure, which is therefore negative, changes continuously from 0 to -r; ..cos decreases continuously from 1 to 0;

.. cos is negative, and changes continuously from 0 to -- 1.

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