Plane trigonometry, by S.L. Loney.

EQUATIONS. 85
87. Ex. 1. Solve the equation 2 sin2x + /3 cos + 1=0.
The equation may be written
2 - 2 cos2 x + ^/3 cos x +1 = 0,
i.e. 2 cos2 x - V/3 cos x - 3 =0,
i.e. (cos x- ^/3) (2 cosx +,/3) = 0.
The equation is therefore satisfied by cos x =/3, or cos= - 2
There is no angle whose cosine is ^/3, so that the first factor gives no
solution.
The smallest positive angle, whose cosine is - 2/, is 150~, i.e. 7r
2 6,/3
Hence the most general value of the angle, whose cosine is - 2
2'
5wr
is 2n7r -. (Art. 83.)
This is the general solution of the given equation.
Ex. 2. Solve the equation tan 50= cot 20.
The equation may be written
tan 5 = tan - 2
Now the most general value of the angle, that has the same tangent as
— 20, is, by Art. 84, n7r+-20,
2 2
where n is any positive or negative integer.
The most general solution of the equation is therefore
50 =nr + - 20
2.. 0= nr+),
where n is any integer.
EXAMPLES. XII.
Solve the equations
1
1 cos2-sin0 —=0. 2. 2sin 2 s +3cos =0.
23 cos=sin 4 cos cos 1.
3. 2V/3 cos2 60=sin 6. 4. cos 6 + cos2 6=1.