Trig equation, is this correct?

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Solve the following equations in the given intervals.

$\displaystyle sec^{2} \theta - ( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 1 $

$\displaystyle 0\leq \theta \leq 2\pi $

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$\displaystyle sec^{2} \theta - ( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 1 $

$\displaystyle 1 + tan^2 \theta = sec^2 \theta $

$\displaystyle 1 + tan^2 \theta - ( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 1 $

$\displaystyle tan^2 \theta -( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 0 $

$\displaystyle tan^2 \theta - tan \theta - \sqrt{3}tan \theta = -\sqrt{3} $

$\displaystyle tan \theta ( tan \theta-\sqrt{3} -1 ) = -\sqrt{3} $

so

$\displaystyle tan \theta = -\sqrt{3} $

or

$\displaystyle tan \theta = 1 $

is this method correct? thanks!

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Originally Posted by Tweety

$\displaystyle sec^{2} \theta - ( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 1 $

$\displaystyle 1 + tan^2 \theta = sec^2 \theta $

$\displaystyle 1 + tan^2 \theta - ( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 1 $

$\displaystyle tan^2 \theta -( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 0 $

$\displaystyle tan^2 \theta - tan \theta - \sqrt{3}tan \theta = -\sqrt{3} $

$\displaystyle tan \theta ( tan \theta-\sqrt{3} -1 ) = -\sqrt{3} $

so

$\displaystyle tan \theta = -\sqrt{3} $

or

$\displaystyle tan \theta = 1 $

is this method correct? thanks!

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it is wrong
let $\displaystyle tan(\theta)=t$ and solve it by $\displaystyle t=\frac{-b\mp \sqrt{b^2-4ac}}{2a}$ in your question a=1 , b=-(1+sqrt(3)) , c=sqrt(3)

then

$\displaystyle tan(\theta)=\frac{-b\mp \sqrt{b^2-4ac}}{2a}$

$\displaystyle \theta = \tan^{-1}\left(\frac{-b\mp \sqrt{b^2-4ac}}{2a}\right)$

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Originally Posted by Amer

it is wrong
let $\displaystyle tan(\theta)=t$ and solve it by $\displaystyle t=\frac{-b\mp \sqrt{b^2-4ac}}{2a}$ in your question a=1 , b=-(1+sqrt(3)) , c=sqrt(3)

then

$\displaystyle tan(\theta)=\frac{-b\mp \sqrt{b^2-4ac}}{2a}$

$\displaystyle \theta = \tan^{-1}\left(\frac{-b\mp \sqrt{b^2-4ac}}{2a}\right)$

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Hi,

I dont understand how you got to the quadratic equation?

the correct answers are, $\displaystyle \frac{\pi}{4} , \frac{\pi}{3} , \frac{5\pi}{4} , \frac{4\pi}{3} $

$\displaystyle tan \theta ( tan \theta-\sqrt{3} -1 ) = -\sqrt{3}
$

You can't factorise when it's like that, you can only factorise when one side is equal to 0

$\displaystyle
tan^2 \theta -( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 0
$

You can use the quadratic equation from there. If it helps let $\displaystyle u = tan \theta$

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Originally Posted by Tweety

Hi,

I dont understand how you got to the quadratic equation?

the correct answers are, $\displaystyle \frac{\pi}{4} , \frac{\pi}{3} , \frac{5\pi}{4} , \frac{4\pi}{3} $

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$\displaystyle tan^2 \theta -( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 0$

let t=tan(theta)

$\displaystyle t^2 -(1+\sqrt{3})t+\sqrt{3}=0$

$\displaystyle t=\frac{1+\sqrt{3}\mp \sqrt{1+2\sqrt{3}+3-4(\sqrt{3})}}{2}$

$\displaystyle t=\frac{1+\sqrt{3}\mp \sqrt{1-2\sqrt{3}+3}}{2}$

$\displaystyle t=\frac{1+\sqrt{3}\mp \sqrt{(1-\sqrt{3})^2}}{2}$

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Originally Posted by Amer

$\displaystyle tan^2 \theta -( 1 + \sqrt{3} )tan \theta + \sqrt{3} = 0$

let t=tan(theta)

$\displaystyle t^2 -(1+\sqrt{3})t+\sqrt{3}=0$

$\displaystyle t=\frac{1+\sqrt{3}\mp \sqrt{1+2\sqrt{3}+3-4(\sqrt{3})}}{2}$

$\displaystyle t=\frac{1+\sqrt{3}\mp \sqrt{1-2\sqrt{3}+3}}{2}$

$\displaystyle t=\frac{1+\sqrt{3}\mp \sqrt{(1-\sqrt{3})^2}}{2}$

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Oh right, I get it now thanks

$\displaystyle \tan^2{\theta} -( 1 + \sqrt{3} )\tan{\theta} + \sqrt{3} = 0 $

$\displaystyle (\tan{\theta} - \sqrt{3})(\tan{\theta} - 1) = 0$

$\displaystyle \tan{\theta} = \sqrt{3}$ ... $\displaystyle \theta = \frac{\pi}{3}$ , $\displaystyle \frac{4\pi}{3}
$

$\displaystyle \tan{\theta} = 1$ ... $\displaystyle \theta = \frac{\pi}{4}$ , $\displaystyle \frac{5\pi}{4}$

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Originally Posted by skeeter

$\displaystyle \tan^2{\theta} -( 1 + \sqrt{3} )\tan{\theta} + \sqrt{3} = 0 $

$\displaystyle (\tan{\theta} - \sqrt{3})(\tan{\theta} - 1) = 0$

$\displaystyle \tan{\theta} = \sqrt{3}$ ... $\displaystyle \theta = \frac{\pi}{3}$ , $\displaystyle \frac{4\pi}{3}
$

$\displaystyle \tan{\theta} = 1$ ... $\displaystyle \theta = \frac{\pi}{4}$ , $\displaystyle \frac{5\pi}{4}$

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how did you get this expression $\displaystyle (\tan{\theta} - \sqrt{3})(\tan{\theta} - 1) = 0$
from
$\displaystyle \tan^2{\theta} -( 1 + \sqrt{3} )\tan{\theta} + \sqrt{3} = 0 $

? is it a difference of two square??

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Originally Posted by Tweety

how did you get this expression $\displaystyle (\tan{\theta} - \sqrt{3})(\tan{\theta} - 1) = 0$
from
$\displaystyle \tan^2{\theta} -( 1 + \sqrt{3} )\tan{\theta} + \sqrt{3} = 0 $

? is it a difference of two square??

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Suppose you had $\displaystyle x^2- (1+ a)x+ a$. Could you see that it factors as (x-1)(x-a)? Do you see that a factors as (-1-)(a) and those add to -(1+ a), the coefficient of x?

Quote:

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Originally Posted by Tweety

how did you get this expression $\displaystyle (\tan{\theta} - \sqrt{3})(\tan{\theta} - 1) = 0$
from
$\displaystyle \tan^2{\theta} -( 1 + \sqrt{3} )\tan{\theta} + \sqrt{3} = 0 $

? is it a difference of two square??

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look at this

$\displaystyle x^2-5x+6 \Rightarrow (x-2)(x-3)$ -2,-3 the multiply of them is 6 and the sum is -5 . 6 is the constant term and -5 is the factor of x

$\displaystyle x^2-4x+3\Rightarrow (x-1)(x-3)$ -1,-3 multiply of them is 3 and the sum is -4 .3 is the constant term and -5 is the factor of x

like this or use

$\displaystyle x=\frac{-b\mp \sqrt{b^2-4ac}}{2a}$ if you do not understand it

Difference of 2 squares here is not needed.

Skeeter has used the fact that $\displaystyle -\sqrt{3}\times -1 = \sqrt{3}$

and that $\displaystyle -\sqrt{3}+ -1 = -(\sqrt{3}+1)$ like what you would do when solving an ordinary quadratic that has 1 as the coeffecient for the $\displaystyle x^2$ term

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Originally Posted by HallsofIvy

Suppose you had $\displaystyle x^2- (1+ a)x+ a$. Could you see that it factors as (x-1)(x-a)? Do you see that a factors as (-1-)(a) and those add to -(1+ a), the coefficient of x?

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if you had the expression $\displaystyle (x-1)(x-a) $ this expands out to $\displaystyle x^2-ax-x +a $

right?

Than you can't really factor it any further to get back to the original expresssion. which is why I can't understand how it turn outs to be (x-1)(x-a)

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Originally Posted by Amer

look at this

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Quote:

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Originally Posted by Amer

$\displaystyle x^2-5x+6 \Rightarrow (x-2)(x-3)$ -2,-3 the multiply of them is 6 and the sum is -5 . 6 is the constant term and -5 is the factor of x

$\displaystyle x^2-4x+3\Rightarrow (x-1)(x-3)$ -1,-3 multiply of them is 3 and the sum is -4 .3 is the constant term and -5 is the factor of x

like this or use

$\displaystyle x=\frac{-b\mp \sqrt{b^2-4ac}}{2a}$ if you do not understand it

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I understand your examples but how does this relate to the trig expression, are you saying that I can use the same method to factorize
$\displaystyle \tan^2{\theta} -( 1 + \sqrt{3} )\tan{\theta} + \sqrt{3} = 0 $ As I would with $\displaystyle x^2-5x+6 $ ?

thanks.

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Originally Posted by Tweety

I understand your examples but how does this relate to the trig expression, are you saying that I can use the same method to factorize
$\displaystyle \tan^2{\theta} -( 1 + \sqrt{3} )\tan{\theta} + \sqrt{3} = 0 $ As I would with $\displaystyle x^2-5x+6 $ ?

thanks.

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yes ...

$\displaystyle x^2 - 5x + 6 = x^2 - 3x - 2x + 6 = x^2 - (3+2)x + 6$

the only difference here is that $\displaystyle 3$ and $\displaystyle 2$ can be combined, wheras the $\displaystyle 1$ and $\displaystyle \sqrt{3}$ cannot.

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Originally Posted by skeeter

yes ...

$\displaystyle x^2 - 5x + 6 = x^2 - 3x - 2x + 6 = x^2 - (3+2)x + 6$

the only difference here is that $\displaystyle 3$ and $\displaystyle 2$ can be combined, wheras the $\displaystyle 1$ and $\displaystyle \sqrt{3}$ cannot.

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oh right finally I get it, that makes so much sense now!!

Thanks!