C H A P T E R 5 Analytic Trigonometry

15.
17.
tan 2x cot x 0 16.
2 tan x
1 tan2 cot x
x
3 tan x cot x
3 tan x cot x 0
3 tan x 1
0
tan x
3 tan2 x 1
tan x
0
1
tan x 3 tan2 x 1 0
cot x3 tan 2 x 1 0
2 tan x cot x1 tan 2 x
2 tan x cot x cot x tan 2 x
2 tan x cot x tan x
cot x 0, 3 or 3 tan 2 x 1 0
x 3
,
2 2 or 3 1 tan2 x 1
3
x 5 7 3 11
, , , , ,
6 2 6 6 2 6
2 sin 2x 0
sin 2x 0
2x n
n
x
2
x 0, 3
, ,
2 2
tan x ± 3
3
sin 4x 2 sin 2x
sin 4x 2 sin 2x 0
2 sin 2x cos 2x 2 sin 2x 0
2 sin 2xcos 2x 1 0
or
x 5 7 11
, , ,
6 6 6 6
cos 2x 1 0
cos 2x 1
2x 2n
Section 5.5 Multiple-Angle and Product-to-Sum Formulas 493
x
n
2
x 3
,
2 2
1
cos x sin x sin2 x 1 sin 2 x 0
sec x 0 or or
No solution
sin x 1
2 sin x 1 0
2
Also, values for which cos x 0 need to be checked.
3
, are solutions.
2 2
sec x2 sin x 1sin x 1 0
x 5 3
, , ,
6 2 6 2
tan 2x 2 cos x 0
sin x
cos x sin2 x
cos x 0
cos x
sin x sin2 x cos 2 x
cos x
2 tan x
1 tan2 2 cos x
x
0
sec x2 sin 2 x sin x 1 0
2 tan x 2 cos x1 tan 2 x
2 tan x 2 cos x 2 cos x tan 2 x
2 tan x 2 cos x 2 cos x sin2 x
cos 2 x
2 tan x 2 cos x 2 sin2 x
cos x
tan x cos x sin2 x
cos x
sin x
cos x cos x sin2 x
cos x
x 5
,
6 6
sin x 1 0
sin x 1
x 3
2