(21.) sin. (+3) - sin. (A - B)=2 sin. ẞ cos.

(20.) sin. (4+3)+sin. (1-3)=2 sin. 0 cos. β

(22.) cos. (+3)+cos. (A - B)=2 cos. A cos. β

(23.) cos. (+3) - cos. (4-3) = - 2 sin. 0 sin. B

(24.) sin. (4+3) cos. (-3)=sin.* -sin.*=cos. - cos. (25.) cos. (+3) cos. (-3)=cos." -sin.*=cos.*4+cos.*ẞ-1 (26.) sin. 6+sin. (+3)+sin. (4+2)+sin. (+33)+ -

sin. (4+nβ)=

sin. (+nß) sin. + (n+1)β

sin.

(27.) cos. +cos. (+3)+cos. (+23)+cos. (+33)+
cos. (+ηβ)
cos. (+) sin. + (n+1) β

..

=

...

sin. B

1

√2

The formulæ of Trigonometry may be multiplied to almost any extent, and the same quantity may be expressed in a vast number of different ways. An intimate acquaintance with those given in the above table is essential to the progress of the student.

The following, although of less frequent occurrence, may occasionally be found useful, and can be readily deduced from the above.

(38.)

sin. (+3)

=

tan. +tan. ẞ cot. cot. A tan. -tan.cot.β-cot.

In order to become familiar with the various combinations, and dexterous in the application of these expressions, the student will do well to exercise himself by verifying the following values of Sin. 6, Cos 4, Tan. 6, which are extracted from the large work of Cagnoli.

6*

2

TABLE OF THE MOST USEFUL ANALYTICAL VALUES OF SIN. 6, COS. 4, ΤΑΝ. Α.

14.

tan. (45°+)+tan. (45) 29. 2cos.(45°+)cos.(45°-)

15. sin. (60°+4)-sin. (60°-4) 30. cos. (60°+4)+cos. (60°-8)

From certain properties of the circles to be discussed in another volume, other important trigonometrical formulæ, may be deduced, furnishing us with more expeditious means of determining, numerically, the values of some of the trigonometrical lines, and ratios, all of which will occur in their order.

To develop sin. x and cos. x in a series ascending by the powers of x.

The series for sin. z must vanish when x=0, and therefore no term in the series can be independent of x, nor can the even powers of x occur in the series; for if we suppose

sin. x =

x+

a1x+a2x2+a3x2 + a4x2 + ax2 + ....

5

then sin. (-x)=-a,x+a2x2-a3x2 + ax-ax2 + ... but sin. (-x)=-sin. x

=-axaxaxaxax

..

a2 =-a, a1=-a,....; hence a2 =0, a1=0....

..