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[Core 3] Addition formulae and identities

The question is:
"Find the exact value of tan(x), given that
sin(x + 30) = 2cos(x - 30)"

Here's what I've got so far:

sin(x) * cos(30) + cos(x) * sin(30) = 2(cos(x) * cos(30) + sin(x) * sin(30))
(sqrt3)/2 * sin(x) + 1/2 * cos(x) = 2((sqrt3)/2 * cos(x) + 1/2 sin(x))

Not sure where to proceed from here, help would be appreciated.
(edited 7 years ago)
Original post by DarkEnergy
The question is:
"Find the exact value of tan(x), given that
sin(x + 30) = 2cos(x - 30)"

Here's what I've got so far:

sin(x) * cos(30) + cos(x) * sin(30) = 2(cos(x) * cos(30) + sin(x) * sin(30))
(sqrt3)/2 * sin(x) + 1/2 * cos(x) = 2((sqrt3)/2 + 1/2 sin(x))

Not sure where to proceed from here, help would be appreciated.


You've lost a cos x from the right hand side.

Then divide each of the 4 terms by cos x and either it will cancel or you'll get sin x / cos x which is tan x.
Reply 2
Avatar for DarkEnergy
DarkEnergy
OP
20
Original post by tiny hobbit
You've lost a cos x from the right hand side.

Then divide each of the 4 terms by cos x and either it will cancel or you'll get sin x / cos x which is tan x.

Alright thanks.

I got

(sqrt3)/2 * tan(x) - tan(x) = sqrt3 - 1/2

What do I do from here?
Original post by DarkEnergy
Alright thanks.

I got

(sqrt3)/2 * tan(x) - tan(x) = sqrt3 - 1/2

What do I do from here?


Take tan x out of the left hand side as a common factor and then divide by the resulting bracket. I'd be inclined to multiply through by 2 first though.
Reply 4
Avatar for DarkEnergy
DarkEnergy
OP
20
Original post by tiny hobbit
Take tan x out of the left hand side as a common factor and then divide by the resulting bracket. I'd be inclined to multiply through by 2 first though.

Thanks, so I got (2 * sqrt(3) - 1) / (sqrt(3) - 2) as my final answer - is this correct?
Reply 5
Avatar for davros
davros
16
Original post by DarkEnergy
Thanks, so I got (2 * sqrt(3) - 1) / (sqrt(3) - 2) as my final answer - is this correct?


I haven't checked your working, but if what you've got there is correct you should be able to rationalize it fairly easily :smile:
Reply 6
Avatar for DarkEnergy
DarkEnergy
OP
20
Original post by davros
I haven't checked your working, but if what you've got there is correct you should be able to rationalize it fairly easily :smile:

Thanks, completely forget about rationalising the denominator - I just multiplied it by sqrt(3) + 2, is that correct? It gave me tan(x) = -4 - 3sqrt(3)
(edited 7 years ago)

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