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# [Core 3] Addition formulae and identities

1. 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.
2. (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.
3. (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?
4. (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.
5. (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?
6. (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
7. (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
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)

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