1991 Question 2b (1 Viewer)

[Paroissien]

A simple question, but difficult to type out.
cosx + (root3)sinx = 1
What method do you use to solve this, and could you please demonstrate the method. My teacher taught me one way, but it takes too damn long

 

[withoutaface]

I've assumed the domain 0<x<2pi
Let t=tan(x/2)
(1-t²)/(1+t²)+2(sqrt3)t/(1+t²)=1
1-t²+2(sqrt3)t=1+t²
0=2t²-2(sqrt3)t
0=t or 0=t-sqrt3

so this means that tan(x/2)=0
x/2=0, pi
x=0, 2pi

or tan(x/2)=sqrt3
x/2=pi/3, 4pi/3
x=2pi/3

x=0, 2pi/3, 2pi

and then you test for x=pi/2, 3pi/2, which give the solutions of sqrt3 and -sqrt3 respectively, and hence dont work

 

[Xayma]

Subsidiary angle.

&radic;3*sin x+cos x=1
Asin(x+&alpha;.)=1
Asin x cos&alpha;+Acos x sin &alpha;=1
Acos &alpha;=&radic;3
Asin &alpha;=1

&there4; tan &alpha;=1/&radic; 3
&alpha;=&pi;/6
A²(cos² &alpha;+sin² &alpha;.)=4
A=2

&there4;
2sin(x+&pi;/6)=1
sin(x+&pi;/6)=1/2
x+&pi;/6=&pi;/6, 5&pi;/6
x=2&pi;(1+3n)/3 or 2n&pi;
OR
x=0, 2&pi;/3 or 2&pi; 0&le;x&le;2&pi;

 

[withoutaface]

Haha, now he has 2 methods

 

[Xayma]

Haha, now he has 2 methods

Well strictly yours is wrong for the domain you specified.

In the domain 0< x <2&pi;

The only answer is 2&pi;/3.

It should be 0&le;x&le;2&pi; Plus it has the added bonus of not needing spaces to keep the x displayed.

OR you could get away with
0-&delta;< x<2&pi;+&delta;

 

[Li0n]

Thanks for beating me to it .

 

[withoutaface]

Well strictly yours is wrong for the domain you specified.

In the domain 0< x <2&pi;

The only answer is 2&pi;/3.

It should be 0&le;x&le;2&pi; Plus it has the added bonus of not needing spaces to keep the x displayed.

You knew what I meant

 

[dawso]

yeah, first method (t formulae) is gud and can be easier wit other harder questions and is needed in 4unit, the auxiliary angle method is gud, once uve learnt the basic rules and method of doin it, you can get out any simple problem, even if u hav no idea on how ur doin it, best and easiest 2 follow, (thats wat xayma used)
-dawso

 

[withoutaface]

t method you must always remember to check pi/2 and 3pi/2

 

[CrashOveride]

Don't sub pi/2 or 3pi/2 back into the equation to check though =p

 

[mojako]

Don't sub pi/2 or 3pi/2 back into the equation to check though =p

??
why? =p(tm)
EDIT: well now I know why =p

(tm) =p(tm) is owned by CrashOveride
or so he claimed..

let's quote CM_Tutor:

There is stuff in the syllabus that does have real life application, and remember that you have to walk before you can run.

hehehe..

 

[paper cup]

t method you must always remember to check pi/2 and 3pi/2

oh thanks for reminding me

 

[gordo]

i reckon the auxillery method is much better than the t method for that kind of question though ey?

 

[CrashOveride]

Because you want to check for say Pi, as it is tan(pi/2) which is not a happy camper

 

[Paroissien]

Subsidiary angle.
2sin(x+&pi;/6)=1
sin(x+&pi;/6)=1/2
x+&pi;/6=&pi;/6, 5&pi;/6
x=2&pi;(1+3n)/3 or 2n&pi;
OR
x=0, 2&pi;/3 or 2&pi; 0&le;x&le;2&pi;

Thanks very much, and this might be a foolish question, but how did you know that there were three answer, when you only had two for x + pi/6 = ...
How do you know when there is a third value?

 

[Xayma]

Well assuming it wants a domain of 0&le;x&le;2&pi;
And you get an answer of 0, then you will also have an answer 2&pi;

since sin 0=sin 2&pi;

 

[tennille]

You have to look at the domain. Im assuming the domain was 0 < x < 2pi
Thne you have to extend the domain because of the pi/6= 30 degrees (so ill just put it in degreed for now)

30 < x + 30 < 390

That means that since sin is positive in the question (sin = 1/2), sin is in the 1st and second quadrants. So there are 2 answers. But the domain has been extended to 390 degrees, so it is in 1st quadrant again as 390 - 30 = 360 degrees = 2pi.

 

[tennille]

i hate when that happens- ur in the process of typing it then someone types it just before u....

 

[Paroissien]

Thanks everyone, that really cleared it up

 

[Li0n]

i hate when that happens- ur in the process of typing it then someone types it just before u....

the second i saw this post i started typing up t-method and auxiliary, then i checked preview post then i see xayma and wof already there...
bastards....