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TABLE OF CONTENTS.
CHAP. I.
DIVISION of circle
..........
Definition of sine; sine of an arc = sine of supplement.....
of versed sine....
of cosine.....
Page 2
3 to 6
Tables of arcs having the same sine and cosine.
16
Tangent, co-tangent, secant, co-secant..
4-16
Radius =r introduced into Trigonometrical Expressions.... 18, &c.
Values of sines, cosines, &c. in particular cases. ....... 20
Table for converting degrees, &c. in the French division
of the circle, into degrees, &c. of the English
division....
23
...
CHAP. II.
Sine, cosine, of an angle of a triangle expressed in terms of
the sides....
24
Sines of angles proportional to sides opposite: area of a
triangle.
26, &c.
1
Expressions for the sine and cosine of the sum of 2 arcs.... 27
sin. (A + B) = sin. A. cos. B + cos. A. sin. B.
cos. (A + B) = cos. A. cos. B + sin. A. sin. B
sin. (A + B) + sin. (A - B) = 2sin. A.cos. B
sin. (A + B) - sin. (A - B) = 2 cos. A. sin. B
sin. (A + B). sin. (A - B) = sin. A - sin. B
cos. (A - B) + cos. (A+B) = 2 cos. A. cos. B
cos. (A - B) - cos. (A+B) = 2 sin. A. sin. B
A + B AB
sin. A + sin. B = 2 sin.
Expression for tan. (A + B + C + &c.)..................... 36
Table of the different expressions for the sine, cosine and
Cos. 5 A= 16 cos.5 A - 20 cos.3 A+ 5 cos. A...
46
sin. 3 A 3 sin. A - 4 sin.3 A.............
sin. 4 A = (4 sin. A - 8 sin.3 A) Cos. A...
sin. 5 A5 sin. A 20 sin. A + 16 sin.5 A....
-
47
Sums of series of the cosines of multiple arcs, of the sines, &c. 50, &c.
Waring's property of chords; Vieta's......... 53
Expressions for the powers of the sine and cosine of an arc
2. cos. A = cos. 2 A + 1...............
...........
22 cos. A = cos. 3 A + 3 cos. A................
23 cos. A = cos. 4 A + 4 cos. 2A + 3......... 60, &c.
24 cos.5 A = cos. 5 A + 5 cos. 3 A + 10. cos. A
25 cos. A = cos. 6 A+6.cos. 4 A +
15 cos. 2 A+ 10.......
2 sin. A = cos. 4A- 4 cos. 2 A+3...............
24 sin.5 A sin. 5 A 5 sin. 3 A+10 sin. A..... 62
Uses of different Solutions of the same case.......... Trigonometrical Problems......
CHAP. VI.
Instances of the utility of Trigonometrical Formulæ.......... 103
CHAP. VII.
Solution of a cubic equation by the Trigonometrical Tables. 118
One side of a spherical triangle < sum of two others.
Three sides of a spherical triangle < great circle.
Method of finding the pole of a great circle..
Supplemental or polar triangle.....
Angles at the base of an isosceles triangle equal.
Greater side opposite the greater angle....... Area of spherical triangle and polygon........
CHAP. IX.
Expression for cosine of angle in spherical triangle...... 139
for sine
......
140
Sines of angles proportional to sines of opposite sides.... 141
CHAP. X.
Formulæ of solution for right-angled spherical triangles... 143
Example of solution of right-angled spherical triangles.. 149
Use of expression for the area of a spherical triangle....
188
Legendre's theorem for solving spherical triangles that are
nearly plane.
191
Reduction of spherical to angles contained by the chords 194
Advantage of Briggs's System of Logarithms.. Explanation of Tables for proportional parts.......
Explanation of the negative Index or characteristic. Expressions for cosine, sine of multiple arc...
235, &c.
Proof of rule for finding sines and tangents of small arcs 254
Sines and tangents computed by differential method.... 257
Legendre's Formula of Reduction......
261
Legendre's Theorem for solving spherical triangles as
rectilinear.
262
..
V