Plane trigonometry with practical applications, by Leonard E. Dickson.

112

TRIGONOMETRY

[Art. 85

triangles T'OX and TOX' have equal hypotenuses and equal
angles at 0, and hence are equal. Thus T'X = TX', OX = OX'.
By the definitions of the trigonometric functions,
sin B = TX' = T'X = sin (180~-B),
cos B = -OX'= - OX = - cos (180 - B).
Also,
sin B sin (1800-B)
tan B — = —tan (180-B).
cos B -cos (180- B)
Whether B is equal to A or to 180~ - A, the three formulas just
proved are the same as the three formulas in the first column of
(1). Taking reciprocals, we get those in the second column; for
example,
1
csc A= ==csc (180-A).
sin A sin (180 -A)csc (180A).
When A = 90~, its tangent and secant are undefined (Art. 83),
while the four formulas (1) which do not involve these two functions are true, since cos 90~ = cot 90~ = 0. 'Hence formulas (1)
hold for every angle between 0~ and 180~ for which the functions
involved are defined.
Second, we shall prove the identities
sin A= -sin (360~-A), csc A = -csc (360~- A),
(2) cos A= cos (360~-A), sec A= sec (360~-A),
tan A = -tan (360~- A), cot A = -cot (360~-A),
which enable us to express the trigonometric functions of any
angle A between 180~ and 360~ in terms of those of 360~ - A,
which is an angle between 0~ and 180~. If the latter angle is
between 90~ and 180~, we saw that, by means of formulas (1) we
can pass to functions of an acute angle, which are given by the
tables.
If A is between 0~ and 180~, write B for 3600 - A. But if A is
between 180~ and 360~, write B for A. In either case, B is an
angle between 180~ and 360~. Place B in its trigonometric position XOT (Fig. 67 if B<270~, Fig. 68 if B> 270~).