damn trig (1 Viewer)

[mcsmells]

Well this is a question out of one of the trial papers we received to practice on...
I was not really sure how to approach it
I asked some of 4 unit buddies except they showed me things i didnt really understand
So here it is...
12cosx + 5sinx = 7
solve for x, when 0 <= x <= 360 degrees

 

[rama_v]

You just use the transformation to convert this into a function involving just sin or cos. In this case:

12cosx + 5sinx = 7

make this into RCos(x-@)
R = sqrt(12² + 5²)

so R = 13
Hence 12cosx + 5sinx = 13Cos(x-@)
= 13CosxCos@ + 13SinxSin@
to find @ just match up both sides, i.e.:
12cosx = 13cosxcos@
12/13 = cos@
@ = 22^o37'

Technically you should do this for Sin@ as well, then use tan@ = sin@/cos@ to verify which quadrant @ lies in. I am presuming that its in the first quadrant in this question but its not always the case.

Anyway, your question now becomes:
13Cos(x- 22^o37') = 7

which is then easy to solve...

 

[speed2]

u could also use the 't-formula'

 

[damo676767]

sub in

sin x = 2 * tan (x/2) / (1 + tan^2(x/2))

and

cos x = (1 - tan^2(x/2)) / (1 + tan^2(x/2))


does anybody rember how you prove them?

 

[香港!]

sub in

sin x = 2 * tan (x/2) / (1 + tan^2(x/2))

and

cos x = (1 - tan^2(x/2)) / (1 + tan^2(x/2))


does anybody rember how you prove them?

tan2x=2tanx\(1-tan²x)
tanx=2tan(x\2)\ (1-tan²(x\2) )

Now by pythagorus:
sinx=2tan(x\2) \ sqrt [ (2tan(x\2))²+(1-tan²(x\2))²]
let t=tan (x\2)
.: sinx=2t\sqrt[4t²+(1-2t²+t^4)]
=2t\sqrt[t^4+2t²+1]
=2t\sqrt[(t²+1)²]
=2t\(1+t²)

Similarly...
cosx=(1-t²)\(1+t²)

 

[damo676767]

i figured that way

just my math treachure showed me a much faster way that i now forget

 

[rama_v]

Yeah, the other way uses triangles with tan (x/2) = t

you also know that sin2@ = 2sin@cos@ so therefore sin@=2sin(@/2)cos(@/2). Just work out the half angles using the triangle and you get the same result

 

[DraconisV]

is this HSC trig, coz the trig that we are doing so far in the prelim is not so advanced as this crazy stuff?

 

[acmilan]

its all prelim trig.

 

[pLuvia]

this is definately preliminary stuff

 

[haboozin]

theres no 3u trig chapter in the HSC course anyways..

so its prelim.

just use t-results and be lazy.
I hate that Rsin/cos(@+Q)

 

[mcsmells]

So what did you guys get as the answer
I used t method and i got an answer but i am not really sure whether it is right as unfortunately i dont have any solutions to the paper

 

[香港!]

So what did you guys get as the answer
I used t method and i got an answer but i am not really sure whether it is right as unfortunately i dont have any solutions to the paper

when you're asked to solve something and you get an answer and ur unsure, can't you just sub ur result back into the equation and see if it holds?

 

[word.]

x = 80°2' and 325°11'