Submultiple angles.

109. Since the relations of Art. 105 are true for all values of the angle A, they will be true if instead of A

A

we substitute and therefore if instead of 2A we put

2'

From equation (2) of the last article we have

2

111. In each of the preceding formulae it will be noted that there is an ambiguous sign. In any particular case the proper sign can be determined as the following examples will shew.

1

Ex. 1. Given cos 45° –

√2

find the values of sin 221° and cos 221⁄2°.

The equation (1) of the last article gives, by putting A equal to 45o,

Now sin 2210 is necessarily positive, so that the upper sign must be

Now 165° lies between 90° and 180°, so that, by Art. 52, its sine is positive and its cosine is negative.

From the above examples it will be seen that, when the angle A and

A its cosine are given, the ratios for the angle may be determined without 2 any ambiguity of sign.

When however only cos A is given, there is an ambiguity in finding The explanation of this ambiguity is given in the next

and cos

article.

Α

**112. To explain why there is ambiguity when cos

2

We know that, if n be any integer,

cos A = cos (2nπ ± A) = k (say).

are found from the value of cos A.

Hence in each case we should expect to obtain two

From equation (1) of Art. 109 we have