Help pls. How to do this?

[peak prairie]

You may take 2 cases to open the modulus

[cold reef]

To solve this question, should I do like this? 1) draw the graph of y=2+3cosx and y=|2+3cosx|, 2)draw y=|2+3sin2x| on the same graph 3) draw the line y=1 and see where it intersects them. Sounds so stupid. 😭

[peak prairie]

that sounds fantastic

but I think for option C there is no necessity to draw the graph
but if you find plotting method more comfortable you may go with it

[cold reef]

hmm, but is it correct...?

[peak prairie]

it surely is

[cold reef]

Thanks so much!

.close

[outer night]

Solved

Post marked as solved by @vast steeple.

Use .unsolved if this was a mistake.

[cold reef]

@peak prairie whats the other method, other than using a graph?

[peak prairie]

for C?

[cold reef]

yes

[peak prairie]

You can open the modulus taking two cases

[cold reef]

yes

[peak prairie]

$2+3sin(2x)=\pm1$

[ember crow]

77²

[peak prairie]

you may solve it as two equations and then take the Union of the cases for full answer

[cold reef]

what do u mean by taking the union of the cases?

[peak prairie]

I mean there could be different values of x satisfying in both cases
since you want the complete set of solutions, you will take union of the answers you get after solving the equations of cases

like if you get Pi, pi/2 from first case and pi/2, 3pi/2 from the second
you will take union which will be pi/2, pi, 3pi/2

[cold reef]

sin 2x = -1/3 and sin 2x = -1 when i solve it, isn't it?

then i should solve further for x i think

[peak prairie]

yeah i think

[peak prairie]

that -1/3 is kinda concerning me

how you are supposed to find the angle

[cold reef]

lemme try that again

[peak prairie]

oh never mind they just want the number of solutions

[peak prairie]

no, you did it correctly

[cold reef]

ok.

should i solve for x further on?

[peak prairie]

it is not necessary
but you just have to work out the number of solutions

[cold reef]

now after doing this, what to do then 😭

[peak prairie]

since $x\in(0,\pi)$ $\implies$ $2x\in(0,2\pi)$

[ember crow]

77²

[peak prairie]

did you get it?

[cold reef]

actually speaking, i just realized that the book has made a mistake for sure, they should be asking about |2 + 3 cos x| = 1 and yes, |2 + 3 sin 2x| = 1 does have 3 intersection in that domain 😭

[peak prairie]

just plot y=1 then

[cold reef]

yes

Thanks @peak prairie alot