Page images
PDF
EPUB

CHAPTER XI.

ON THE TRIGONOMETRICAL RATIOS FOR MULTIPLE AND

SUBMULTIPLE ANGLES.

123. THE angles 24, 34, 44 ...... are called Multiples of 4, A A A

and the angles,..... are called Submultiples of A.

We shall shew how to express the Trigonometrical Ratios of 2A, 3A and

A

2

in terms of the ratios of A,

[blocks in formation]

Now we may put 1-sin3 A in the place of cos3 A, (Art. 84) and we then have

cos 24=1-sin2 A - sin' A

= 1-2 sin A.

Or we may put 1-cos A in the place of sin A, (Art. 84) and we then have

[merged small][ocr errors][merged small][merged small][merged small][merged small]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

= (2 sin A. cos A). cos A + (1 − 2 sin3A). sin A

=

--

[blocks in formation]
[blocks in formation]

128. Tan 34

= cos 24. cos A-sin 24. sin A

=3

= (2 cos3 A − 1). cos A – (2 sin A. cos A). sin A

[blocks in formation]
[blocks in formation]

In some of these Examples the student may employ with advantage the formulæ given in Art. 116.

[blocks in formation]
« PreviousContinue »