Derzhavin's Rectilinear Trigonometry
by S. S. Derzhavin
Translated by Valery Manokhin, PhD
A complete classical course in plane trigonometry, in first complete English translation. Eight divisions and 86 numbered sections run from the function concept and radian measure through the trigonometric functions traced quadrant by quadrant, the addition and reduction formulas, the construction and use of logarithmic tables, the solution of oblique triangles by the laws of sines and cosines and by Mollweide’s formulas, measurement in the field and triangulation, the computation of π, and trigonometric equations. Sixty-eight figures and Tables I–V.
8 chapters · 86 numbered articles · 192 pages
Where this fits in school — US, UK and Canada ↓ Adults and self-learners welcome — great mathematics has no age limit.
Where this fits in school
Full curriculum table — every book by school year →| United States | 10–12 | Trigonometry / Precalculus; the field-measurement chapter has no modern counterpart |
|---|---|---|
| United Kingdom | Years 10–13 | GCSE Higher (sine and cosine rules) → A-level Pure |
| Canada | Ontario 10–12 | MPM2D → MCR3U → MHF4U · QC Secondary IV–V SN |
Not covered in this book
statistics and data handling · probability · vectors
Source these strands elsewhere. They are missing from the book, not from this description.
How this book fits
See the whole sequence →Work through first
- Elementary Algebra$69.95
- Kiselev's Algebra II$54.95
Covers much of the same ground
Nearly the same length — 188 pages to 192 — but Anoshchenko gives 57 pages to supplementary articles: trigonometric equations, inverse trigonometric quantities, maxima and minima. This book gives those six numbered sections, and adds the derivative, the series for sine and cosine, the computation of π and Tables I–V, none of which Anoshchenko carries.
- Rashevsky's Trigonometry$39.95
Rashevsky is the short course — 81 pages to 192. It carries no logarithmic tables, no chapter on measurement in the field and no computation of π.
These overlap heavily. Each note says what that book has which this one does not — the differences are what decide between them.
Problems to work alongside it
What comes next
- Kiselev's Calculus$54.95
Table of contents
93 section headings — the printed contents page in full.
Translator's Preface
Introduction6 sections
- §1. The concept of a function
- §2. Continuous change of the argument and of the function
- §3. Geometric representation of functions
- §4. The concept of the derivative
- §5. Mechanical meaning of the derivative
- §6. Measurement of angles in radians
The concept of trigonometric functions and their variation19 sections
- §7. Equation of a circle with center at the origin
- §8. The concept of the functions sin t and cos t
- §9. Positive and negative direction of arcs
- §10. Variation of the function sin t in connection with the variation of the arc from 0 to ± 2kπ
- §11. Table of values of the function sin t
- §12. Graphical representation of the variation of the function sin t
- §13. Variation of the function cos t in connection with the variation of the arc from 0 to ± 2 k π
- §14. Graphical representation of the variation of the function cos t
- §15. The concept of the functions tg t and ctg t
- §16. Variation of the function tg t in connection with the variation of the arc from 0 to ± kπ
- §17. Graphical representation of the variation of the function tg t
- §18. Variation of the function ctg t in connection with the variation of the arc from 0 to ± kπ
- §19. Graphical representation of the variation of the function ctg t
- §20. Expression of tg t and ctg t as functions of sin t and cos t
- §21. The concept of the functions sc t and csc t
- §22. Variation of the functions sc t and csc t in connection with the variation of the arc from 0 to ± 2kπ
- §23. Formulas relating sc t and csc t to the other trigonometric functions
- §24. Relations between the elements of a right triangle
- §25. Solution of right triangles
Transformation of coordinates and identical trigonometric transformations22 sections
- §26. Relations between the trigonometric functions of the arcs +t and -t
- §27. Transformation of coordinates
- §28. Reduction formulas for sine and cosine
- §28a. A second method of deriving the reduction formulas for sine and cosine
- §29. Reduction formulas for tangent, cotangent, secant, and cosecant
- §30. The concept of circular functions
- §31. Properties of the circular functions
- §32. General expression for arcs having the same sine or cosecant
- §33. General expression for arcs having the same cosine or secant
- §34. General expression for arcs having the same tangent or cotangent
- §35. Examples
- §36. Transformation of the direction of the coordinate axes
- §37. Sine and cosine of the sum of two arcs
- §38. Sine and cosine of the difference of two arcs
- §39. Tangent of the sum and difference of arcs
- §40. Theorem
- §41. A second method of obtaining the formulas for the sine and cosine of the sum and difference of two arcs
- §42. Generalization of the formulas for the sine and cosine of the sum and difference of two arcs
- §43. Sine, cosine and tangent of double arcs
- §44. Sine, cosine and tangent of half an arc
- §45. Reduction of expressions to a form convenient for logarithmic computations
- §46. Reduction of expressions to logarithmic form by introducing an auxiliary angle
Tables of values of trigonometric functions and their logarithms14 sections
- §47. Theorems on which the computation of values of trigonometric functions for small angles is based
- §48. Approximate computation of sin 0.01 and cos 0.01
- §49. Approximate calculation of sin 1' and cos 1'
- §50. Simpson's formulas; their application to the construction of tables of values of trigonometric functions
- §51. Expansion of sin t and cos t into infinite series
- §52. Differences of successive values of sines and cosines
- §53. Tables of logarithms of trigonometric quantities
- §54. Finding the logarithm of a trigonometric function of a given angle
- §55. Theorem
- §56. Finding the logarithms of the sine and tangent for angles close to zero, and also the logarithms of the cosine and cotangent for angles close to a right angle
- §57. Finding an angle from the logarithm of a trigonometric function
- §58. Finding an angle from the logarithm of a trigonometric function in the case of a large tabular difference
- §59. Derivatives of trigonometric functions
- §60. Simple harmonic motion
Solution of oblique triangles12 sections
- §61. Theorem. The sum of two sides of a triangle is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half the difference of the same angles
- §62. Mollweide's formulas
- §63. Theorem
- §64. Expression of the trigonometric functions of the angles of an oblique triangle in terms of its sides
- §65. Area of a triangle
- §66. Theorem
- §66a. Corollaries
- §67. Solution of oblique triangles
- §68. Numerical examples on the solution of triangles
- §69. Examples of more complicated cases of solving oblique triangles
- §70. Problem
- §71. Solution of regular polygons
Measurements on the ground9 sections
- §72. Surveying; vertical and horizontal directions
- §73. Designation and measurement of lines on the ground
- §74. Measurement of angles
- §75. Laying out on the ground mutually perpendicular and parallel lines with the aid of a cross-staff
- §76. Determination of the relative heights of two points
- §77. Application of trigonometry to the performance of various measurements on the ground
- §78. Determination of heights
- §79. Determination of inaccessible distances
- §80. Triangulation
Computation of π3 sections
- §81. Addition and subtraction of circular functions
- §82. Expansion of arc tg x in powers of x
- §83. Computation of π
Trigonometric equations3 sections
- §84. General remarks
- §85. Examples of trigonometric equations with one unknown
- §86. Examples of solving systems of trigonometric equations with two unknowns
Tables5 sections
- Table I
- Table II
- Table III
- Table IV
- Table V
Formats and editions
| Format | Price | ISBN-13 | Where | Buy |
|---|---|---|---|---|
| Paperback | $54.95 | 978-1-80910-009-2 | Amazon | Buy → |
| Hardcover | $84.95 | 978-1-80910-010-8 | Amazon | Buy → |
Prices are the publisher list prices. Amazon sets the final price in each store.
Product details
- Publisher
- Northern Star Academic Press
- Language
- English
- Translator
- Valery Manokhin, PhD
- Print length
- 192 pages
- First published
- 2026-09-16
Published by Northern Star Academic Press · First faithful English translation from the original Russian edition
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