Plane trigonometry, by S.L. Loney.

276

TRIGONOMETRY.

In the above we have tacitly assumed that ab < 1, so that is posi1 - Iab
tive, and therefore tan-l 1 b lies between 0~ and 90~.
1- ab
a+b
If however abbe> 1, then 1 —and therefore according to our defini1-ab
a+b
tion tan-'1 - is a negative angle. Here y is therefore a negative angle
1- ab
and, since tan (r + y) = tan y, the formula should be
a+b
tan-l a + tan-' b = w + tan- a
1 -ab
Ex. 6. Solve the equation
tan- x- + tan-l = tan-l (- 7).
x
Here we have
a+=/3=,
x+l x+l
where a = tan-l1 and hence tan a = +,
x- 1 xi-1
= tan1 -- and hence tan = -
sc x
and y = tan-i (- 7) and hence tan = - 7.
Since tan (a + ) = tan y,
tan a + tan _
1-tanatan p
x+l x-1
x-1 x
i. e.. -.7,
1,.. 1 ~x+l x-1 1'
x-1 x
i. e.2X Xl 71
u-x+l
i.e. 1- =-7'
so that x=2.
This value makes the left-hand side of the given equation positive, so
that there is no value of x strictly satisfying the given equation.
The value x= 2 is a solution of the equation
tan-1 X+ 1 + tan-1 -1 = i r tan-1 ( - 7).
x - 1 x