Question 960547: Find all solutions if
0 ≤ x < 2π.
Use exact values only. (Enter your answers as a comma-separated list.)
cos 2x cos x − sin 2x sin x = square root of 2 over 2
x=?
Found 2 solutions by lwsshak3, ikleyn:
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Find all solutions if
0 ≤ x < 2π.
Use exact values only. (Enter your answers as a comma-separated list.)
cos 2x cos x − sin 2x sin x = square root of 2 over 2
x=?
***
Identity: cos(x+y)=cosxcosy-sinxsiny
cos 2x cos x − sin 2x sin x = square root of 2 over 2
cos(2x+x)=√2/2
2x+x=π/4, 7π/4
3x=π/4, 7π/4
x=π/12, 7π/12
Answer by ikleyn(53937)  (Show Source):
You can put this solution on YOUR website! .
Find all solutions in 0 ≤ x < 2π.
Use exact values only. (Enter your answers as a comma-separated list.)
cos 2x cos x − sin 2x sin x = square root of 2 over 2
x=?
~~~~~~~~~~~~~~~~~~~~~~~~~~~
In his post, @lwsshar3 found two solutions,  and  .
Actually, this equation has 6 (six) solutions, but @lwsshar3 missed/lost most of them.
See my correct and complete solution below.
Use identity: cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)
cos(2x)* cos(x) − sin(2x)* sin( x) = square root of 2 over 2
cos(2x+x) = √2/2
2x+x = π/4 + 2k*π; 7π/4 + 2k*π.
3x = π/4 + 2k*π; 7π/4 + 2k*π.
For 3x, you must add the periods 2π and 4π for cosine.
For 3x, it will produce geometrically the same angle;
but for 'x' it will produce new angles that you will miss otherwise.
You should no add more hire periods for 3x, since it will not produce
new angles for x and will lead you out of the given interval.
Now consider two cases.
Case (a). 3x =  ,  ,  .
then x = ,  =  =  ,  =  .
Case (b). 3x =  ,  ,  .
then x = ,  =  =  ,  =  .
ANSWER. The given equation has 6 solutions in the given interval
, , , , , .
Solved completely and correctly - no one root is missed.
This analysis in my post is typical in trigonometry problems, when you work with a multiple of an angle.
If you miss this analysis, you will miss many solutions to your original equation.
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