+1124 Hi there good people!,
I have not been on here for almost 2 years!!..golly,..time flies!!..I trust you are all doing great. Please help me solve for X using the following problem:
(CosX + 2SinX)(3Sin2X-1)=0
I know the answers must be x=153,43 deg +k.180 OR x=153,43 deg +k.360 OR x=333,43 deg +k.360
AND
x=9,74 deg +k.180 OR x=180,26 deg +k.180.
Please help me out here, any and all help will be greatly appreciated!..Thank you all.
+118673
Hi Juriemajic,
Good to see you again.
IGNORE THIS ANSWER, i HAVE REDONE IT.
(CosX + 2SinX)(3Sin2X-1)=0
If the product of two terms is 0 then one or the other or both must equal zero.
so
(1) (CosX + 2SinX)=0 or (2) (3sin 2x-1)=0
(1) \(cos\; x=-2sin\;x\\ -0.5 = tan\;x \\ \text{2nd or 3rd quad}\\ x = 180-atan(0.5)+360n \qquad or \qquad x=180+atan(0.5)+360n\\ x = 180-26.565+360n \qquad or \qquad x=180+26.565+360n\\ x = 153.434+360n \qquad or \qquad x=206.565+360n\\ \)
(2) \(3Sin\;2x-1=0\\ sin\;2x=\frac{1}{3}\\ \text{1st or 2nd quad}\\ \qquad asin\;\frac{1}{3}=19.471^o\\ \\~\\ 2x=19.471+360n \qquad or \qquad 2x=180-19.471+360n\\ 2x=19.471+360n \qquad or \qquad 2x=160.529+360n\\ x=9.736+180n \qquad or \qquad x=80.265+180n\\ \)
+1124 Hewllo sweet Melody!!,
Thank you for replying....Melody, may I ask why we always use the "+360n" thingy????
Also, the answers in the memorandum indicates:x=153,43 deg +k.180
OR x=153,43 deg +k.360
OR x=333,43 deg +k.360 for (CosX + 2SinX)=0
and x=9,74 deg +k.180
OR x=180,26 deg +k.180. for (3sin 2x-1)=0
Also, you have added values..(more answers)....This is something I truly do not grasp...please will you explain for dummies? 
+118673 Hi Juriemajic,
I made some errors, I will fix them in red.
I found one of my first answer was erraneous but for the second set of answers your given answers were not both correct.
(CosX + 2SinX)(3Sin2X-1)=0
If the product of two terms is 0 then one or the other or both must equal zero.
so
(1) (CosX + 2SinX)=0 or (2) (3sin 2x-1)=0
(1)
\(cos\; x=-2sin\;x\\ -0.5 = tan\;x \\ \text{2nd }or \textcolor{red}{\;\;4th\;\; quad}\\ x = 180-atan(0.5)+360n \qquad or \qquad \textcolor{red}{x=360-atan(0.5)+360n}\\ x = 180-26.565+360n \qquad or \qquad \textcolor{red}{x=-26.565+360n}\\ \text{combining them I get}\\ \textcolor{red}{ x=-26.565+180n}\)
I graphed this to check it was correct
I didn't find any mistakes in my second answer and I have verified it with the graph.
\(3Sin\;2x-1=0\\ sin\;2x=\frac{1}{3}\\ \text{1st or 2nd quad}\\ \qquad asin\;\frac{1}{3}=19.471^o\\ \\~\\ 2x=19.471+360n \qquad or \qquad 2x=180-19.471+360n\\ 2x=19.471+360n \qquad or \qquad 2x=160.529+360n\\ x=9.736+180n \qquad or \qquad x=80.265+180n\\ \)
Latex 1
cos\; x=-2sin\;x\\ -0.5 = tan\;x \\
\text{2nd }or \textcolor{red}{\;\;4th\;\; quad}\\
x = 180-atan(0.5)+360n \qquad or \qquad \textcolor{red}{x=360-atan(0.5)+360n}\\
x = 180-26.565+360n \qquad or \qquad \textcolor{red}{x=-26.565+360n}\\
\text{combining them I get}\\
\textcolor{red}{ x=-26.565+180n}
Latex2
3Sin\;2x-1=0\\ sin\;2x=\frac{1}{3}\\ \text{1st or 2nd quad}\\ \qquad asin\;\frac{1}{3}=19.471^o\\ \\~\\ 2x=19.471+360n \qquad or \qquad 2x=180-19.471+360n\\ 2x=19.471+360n \qquad or \qquad 2x=160.529+360n\\ x=9.736+180n \qquad or \qquad x=80.265+180n\\
+118673 Our answers for cosx+2sinx=0 are now the same becasue
333.43-360=-26.57
153.43-180=-26.57
Now, you say you do not understand.
I would like more guidence before I start explaining more.
Can you get the answers without the +180k or +360k part.
That is the first thing you need to be able to do.
Maybe solve for just one answer first,
then try and answer for all answers between 0 and 360 degrees. (usually just 2 answers)
Can you solve for 1 answer or are you stumped right from the beginning.?
+1124 Good morning Melody,
You are such a great help to me, thank you so so much..
I have gone through the answers very SLOWLY, taking in and seeing the cartesian plane, and I have suddenly realized why we start of with
\(x=360-\) or \(x=360+\) or \(x=180 +/-\)
because it depends which quads you work with....Finally this has become clear...I am only left with the last parts which say \(+/- 360n/180n\)
If only that can be explained please. I have a hunge that it may have to do with limits?..or restrictions?
+118673 I do suggest you watch the video that I put up on another or your posts today.
the plus/minus 360 is easy
sin of any angle is equal to the sin of that angle plus or minus any multiple of 360degrees
so
sin23 = sin(23+360) = sin (23 - 720) = sin (23 +/- 360n) Where n is an integer
same goes for any other trig function.
with tan we can go one step further
tan is positive in the 1st and third quadrants.
so
\(tan\theta=tan(180+\theta) =tan (360+\theta) \qquad \text{where theta is an acute angle}\\ also\\ tan(180-\theta)=tan(360-\theta) \\ \\this \;means\;that\;\\ \text{We know that }tan60^o=\frac{\sqrt3}{2}\\ so\;we \;know\\ atan\frac{\sqrt3}{2}=60\\ \text{but it also equals} \;\;60+180, 60-720,etc, \quad 60+180n\\ atan\frac{\sqrt3}{2}=60\pm180n\\ \text{We only need the plus becasue n is any integer, it can be negative.}\\ atan\frac{\sqrt3}{2}=60+180n\;\;\;degrees\\\)
I hope I haven't made any stupid mistakes 
LaTex:
tan\theta=tan(180+\theta) =tan (360+\theta) \qquad \text{where theta is an acute angle}\\
also\\ tan(180-\theta)=tan(360-\theta) \\
\\this \;means\;that\;\\
\text{We know that }tan60^o=\frac{\sqrt3}{2}\\
so\;we \;know\\
atan\frac{\sqrt3}{2}=60\\
\text{but it also equals} \;\;60+180, 60-720,etc, \quad 60+180n\\
atan\frac{\sqrt3}{2}=60\pm180n\\
\text{We only need the plus becasue n is any integer, it can be negative.}\\
atan\frac{\sqrt3}{2}=60+180n\;\;\;degrees\\
