Download tan 2x=8cos2 x−cot x

6-5 Solving Trigonometric Equations
To solve trigonometric equations, you can use the following strategies:
1. As necessary, use the double angle formulae to match the arguments
2. As necessary, use formulae to write the equation in terms of one trigonometric function
3. Use algebraic methods to solve for the trigonometric function involved.
4. Solve for the argument
5. If the argument is other than a single variable, solve for the single variable
6. Find all the solutions within the specified domain.
Example 1 Solve for x:


tan 2 x=8cos2 x−cot x ,− ≤x≤
2
2
Solution:
tan 2 x=8cos2 x−cot x
2 tan x
=8cos 2 x −cot x
2
1−tan x
2 tan x=8cos2 x−cot x 1− tan 2 x 
2 tan x=8cos2 x−cot x−8sin 2 tan x
2 sin x
cos x
sin x
=8cos2 x−
−8sin x
cos x
sin x
cos x
sin x cos x

=8cos2 x−sin 2 x 
cos x sin x
sin 2 xcos2 x
=8cos 2 x −sin 2 x 
sin x cos x
2
2
1=8cos x−sin x sin x cos x 
1=4 cos 2 x sin 2 x
1=2sin 4 x
1
=sin 4 x
2
 5
thus 4 x= ,
6 6
 5
and x = ,
24 24
Example 2: Solve for x:
2sin 4 xcos4 x=1,−≤x≤
Solution:
2 sin 4 x2 sin 2 x cos 2 xcos4 x −4 sin 2 x cos2 x=1
2sin 2 xcos 2 x 2 −4 sin 2 x cos 2 x =1
2−4sin 2 x cos2 x=1
−4 sin 2 x cos2 x=−1
1
sin 2 x cos2 x=
4
1
2
2
sin x 1−sin x=
4
1
sin 2 x−sin 4 x− =0
4
4
2
4 sin x−4 sin x1=0
2
2
2 sin x−12sin x−1=0
1
2
sin x=
2
±1
sin x=
2
3   3 
so x=−
,− , ,
4
4 4 4
Also study the examples in the textbook.
Assignment: Page 291 # 1, 2, 3 a, c, e, 4 d, f