Periodicity in trig equations

[tawny helm]

If we solve a trig equation, say $cos(3x) = \frac{\sqrt{3}}{2}$, which can be calculated as $3x = \frac{\pi}{6}$, when should I add $2\pi k$ to the answer? Adding a term $2\pi k$ results in the same angle as the angle does a full rotation, so why does it matter if I add this term before or after dividing the above equation by 3? My textbook states that one should add it first so the answer would be $x = \frac{\pi}{12} + \frac{2\pi k}{3}$

[warm hazel]

Pen

[tawny helm]

But, adding 2pik after dividing by 3 should yield in an equally correct angle as well

Is the reason of why we add the term before dividing since one can think of the term always being there from the start?

[oblique salmon]

If you divide by 3 you should end up with pi/18

but you can add 2pik after wards as well

[tawny helm]

My bad, you are correct

[tawny helm]

That's what I'm thinking

So, why do most answers prefer to add 2pik before dividing?

[oblique salmon]

it's just when you are asked for all solutions

you add 2pi k

[tawny helm]

Yeah, when you want all solutions

[oblique salmon]

because alls solutions are multiples of 2pi away

[tawny helm]

Yes, I do understand that

If we go back to the example, 3x = pi/6, why do we first add 2pik then divide, isntead of adding 2pik after dividing by 3?

Both are correct, right?

Since, if you add 2pik before dividing, it's the same as not adding anything

[oblique salmon]

oh wait

my bad

if we add 2pi afterwards

we are skipping solutions

the thing is if you noticed

the period of cos(3x)

it's 2pi/3

hence each solutions is 2pi/3 away

thats why 2pi is being added before

[tawny helm]

so we'd be skipping 2/3 of the solutions?

[oblique salmon]

yea basically

[tawny helm]

and we can do this since any angle x, x+ 2pi is the same as x + 0 ?

I didn't realize we'd be skipping the solutions if we did add after

Now I understand why we do it before dividing

[oblique salmon]

i mean you are adding a bigger period

[tawny helm]

Yeah, that makes it more clear on the total solutions

Thank you so much for your time and patience, Adonis! I finally understand this now : )

[oblique salmon]

no problem

[latent moth]

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[agile sky]

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