Question

Find all the values of x in the interval [0, 2pie] that satisfy the equation 2cos(x) + sin(2x) = 0.

Answer 1

step one is to rewrite \[sin(2x)=2sin(x)cos(x)\]

Answer 2

then you get \[2cos(x)+2cos(x)sin(x)=0\]
\[2cos(x)(1+sin(x))=0\]
\[cos(x)=0\] or
\[1+sin(x)=0\]
\[sin(x)=-\frac{1}{2}\]

Answer 3

can you solve from there?

Answer 4

can you continue please?

Answer 5

sure. you are in the interval \[(0,2\pi)\]

Answer 6

in that interval cosine is 0 at \[\frac{\pi}{2}\] and \[\frac{3\pi}{2}\]

Answer 7

are those fractions?

Answer 8

if you do not instantly know where
\[sin(x)=-\frac{1}{2}\] then look at the cheat sheet
http://tutorial.math.lamar.edu/cheat_table.aspx
and see that it is at \[\frac{7\pi}{6}\] and \[\frac{11\pi}{6}\]

Answer 9

that is where the second coordinate is \[-\frac{1}{2}\]

Answer 10

unit circle on last page of cheat sheet