CHAPTER XI

ON THE TRIGONOMETRICAL RATIOS OF MULTIPLE ANGLES

AND SUBMULTIPLE ANGLES

88. To express the trigonometrical ratios of the angle 2 A in terms of those of the angle A.

.

Since sin (A+B)= sin A ⋅ cos B + cos A · sin B;

.. sin (A+ A)= sin A⋅ cos A+ cos A · sin A;

.. cos (A + A)= cos A⋅ cos A - sin A. sin A;

91. SUBMULTIPLE ANGLES. In formulæ (v.) the angle A is any angle. Hence we may write A=

The formulæ (v.) now become

sin a= 2 sin

α

2

*92. An examination of formulæ (v.) shows that if sin A or cos A be given, cos 2 A is uniquely determined. The converse is not true; i.e. if cos 2 A is given, sin A and cos A have a sign ambiguity.

tan A

sin A

cos 2 A

=

2

This presents a fourfold sign ambiguity.

Similar remarks apply with equal force to the submultiple angles.

NOTE. The similarity of these two results is likely to cause confusion. This may be avoided by observing that the second formula must be true when A=0°; and then cos 3 A= cos 0° 1. In which case the formula gives cos 0° = 4 cos 0° - 3 cos 0°, or 1 = 43, which is true. The first formula may be proved thus:

=

sin 3 A = sin (2 A + A) = sin 2 A⋅ cos A+ cos 2 A · sin A

The second formula may be proved in a similar manner.