Plane trigonometry, by S.L. Loney.

*72

TRIGONOMETRY.

73. To find the trigonometrical ratios of (180~ + 0) in
terms of those of 0, for all values of 0.
The required relations may be obtained geometrically,
as in the previous articles. The figures for this proposition are easily obtained and are left as an example for
the student.
They may also be deduced from the results of Art. 70,
which have been proved true for all angles. For putting
90 + 0 = B, we have
sin ( 180~ + )= sin (90~ + B) = cos B (Art. 70)
= cos (90~ + )= - sin 0, (Art. 70)
and cos (180~ + O )=cos (90 + B) =- sin B (Art. 70)
= - sin (90 + 0)= - cos 0. (Art. 70).
So tan (180~ + 0)= tan (90~ + B)= -cot B
= -cot (90~ + 0) = tan 0,
and similarly cot (180~ + 0) = cot 0,
see (180~ + 0) =-see 0,
and cosec (180 + 0) =- cosec 0.
74. To find the trigonometrical ratios of an angle
(360~ + 0) in terms of those of 0, for all values of 0.
In whatever position the revolving line may be when
it has described any angle 0, it will be in exactly the same
position when it has made one more complete revolution
in the positive direction, i.e. when it has described an
angle 360~ + 0.