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Thread: when does cos^2x=sin^2x?

  1. #1
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    when does cos^2x=sin^2x?

    when does cos^2 x=sin^2 x?
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  2. #2
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    Quote Originally Posted by Amberosia32 View Post
    when does cos^2 x=sin^2 x?
    You should know from the Pythagorean Identity that

    $\displaystyle \cos^2{x} + \sin^2{x} = 1$.

    So $\displaystyle \cos^2{x} = 1 - \sin^2{x}$.


    Substituting this into your original equation:

    $\displaystyle \cos^2{x} = \sin^2{x}$

    $\displaystyle 1 - \sin^2{x} = \sin^2{x}$

    $\displaystyle 1 = 2\sin^2{x}$

    $\displaystyle \sin^2{x} = \frac{1}{2}$

    $\displaystyle \sin{x} = \pm\frac{1}{\sqrt{2}}$

    $\displaystyle x = \left \{ \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \right \} + 2\pi n$, where $\displaystyle n$ is an integer representing the number of times you have gone around the unit circle.
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    Quote Originally Posted by Amberosia32 View Post
    when does cos^2 x=sin^2 x?
    $\displaystyle Cos^2x=Sin^2x$

    $\displaystyle Cos^2x-Sin^2x=0$

    $\displaystyle \left(Cosx+Sinx\right)\left(Cosx-Sinx\right)=0$

    $\displaystyle Cosx=Sinx,\ or\ Cosx=-Sinx$

    As Cosx gives the horizontal co-ordinate and Sinx gives the vertical co-ordinate of the unit circle centred at the origin,

    Cosx=Sinx at $\displaystyle \frac{\pi}{4}+2n\pi,\ \left(\pi+\frac{\pi}{4}+2n\pi\right)$ for n=0,1,2.....

    Cosx= -Sinx at $\displaystyle \left(\pi-\frac{\pi}{4}+2n\pi\right),\ \left(2{\pi}-\frac{\pi}{4}+2n\pi\right) $ for n=0.1.2....
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    Hello, Amberosia32!

    When does $\displaystyle \cos^2\!x \:=\:\sin^2\!x\;?$

    We have: .$\displaystyle \sin^2\!x \:=\:\cos^2\!x$

    Divide by $\displaystyle \cos^2\!x\!:\;\;\;\frac{\sin^2\!x}{\cos^2\!x} \:=\:1 \quad\Rightarrow\quad \left(\frac{\sin x}{\cos x}\right)^2 \:=\:1 \quad\Rightarrow\quad \tan^2\!x \:=\:1$ . $\displaystyle \Rightarrow\quad \tan x \:=\:\pm1$

    Therefore: .$\displaystyle x \;=\;\frac{\pi}{4} + \frac{\pi}{2}n\;\;\text{ for any integer }n$

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  5. #5
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    cos^2 x = sin^2 x?
    Attached Thumbnails Attached Thumbnails when does cos^2x=sin^2x?-ikt2.gif   when does cos^2x=sin^2x?-ikt.gif  
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