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Thread: arccos+arccot

  1. #1
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    arccos+arccot

    Calculate . The answer may not contain cyklometriska functions.

    got no ide to do this
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    Re: arccos+arccot

    Hello, Petrus!

    $\displaystyle \text{Calculate: }\:\theta \;=\;\underbrace{\arccos\left(\tfrac{3}{5}\right)} _{\alpha} + \underbrace{\text{arccot}\left(\tfrac{1}{7}\right) }_{\beta}$

    Let $\displaystyle \alpha \,=\,\arccos\left(\tfrac{3}{5}\right) \quad\Rightarrow\quad \cos\alpha \,=\,\tfrac{3}{5} \,=\,\tfrac{adj}{hyp}$
    . . Then: .$\displaystyle opp \,=\,4\quad\Rightarrow\quad \tan\alpha \,=\,\tfrac{4}{3}$

    Let $\displaystyle \beta \,=\,\text{arccot}\left(\tfrac{1}{7}\right) \quad\Rightarrow\quad \cot\beta \,=\,\tfrac{1}{7}$
    . . Then: .$\displaystyle \tan\beta \,=\,\tfrac{7}{1} \,=\,7$


    We have: .$\displaystyle \theta \;=\;\alpha + \beta$

    . . $\displaystyle \tan\theta \;=\;\frac{\tan\alpha + \tan\beta}{1 - (\tan\alpha)(\tan\beta)} \;=\;\frac{\frac{4}{3}+7}{1-\frac{4}{3}(7)} \;=\;\frac{\frac{25}{3}}{\text{-}\frac{25}{3}} \;=\;-1 $


    $\displaystyle \text{Therefore: }\:\theta \;=\;\frac{3\pi}{4} + \pi n\;\text{ for any integer }n.$

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  3. #3
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    Re: arccos+arccot

    Let $\displaystyle \mbox{arc}\cos\left(\frac{3}{5}\right)+\mbox{arc} \cot\left(\frac{1}{7}\right)=x$
    Now, let's take the $\displaystyle \cos$ of both sides. Thus
    $\displaystyle \cos\left[\mbox{arc}\cos\left(\frac{3}{5}\right)+\mbox{arc} \cot\left(\frac{1}{7}\right)\right]=\cos(x)$
    $\displaystyle \Leftrightarrow \cos\left[\mbox{arc}\cos\left(\frac{3}{5}\right)\right]\cos\left[\mbox{arc} \cot\left(\frac{1}{7}\right)\right]-\sin\left[\mbox{arc}\cos\left(\frac{3}{5}\right)\right]\sin\left[\mbox{arc} \cot\left(\frac{1}{7}\right)\right]=\cos(x)$

    If we use the following identities:
    $\displaystyle \cos\left[\mbox{arc}\cos(x)\right]=x$
    $\displaystyle \cos\left[\mbox{arc}\cot(x)\right]=\frac{1}{\sqrt{\left(\frac{1}{x}\right)^2+1}}$
    $\displaystyle \sin\left[\mbox{arc}\cos(x)\right]=\sqrt{1-x^2}$
    $\displaystyle \sin\left[\mbox{arc}\cot(x)\right] =\frac{1}{\sqrt{1+x^2}}$

    then we can write the equation as
    $\displaystyle \left(\frac{3}{5}\right)\frac{1}{\sqrt{50}}-\sqrt{1-\left(\frac{3}{5}\right)^2}\frac{1}{\sqrt{1+\left( \frac{1}{7}\right)^2}}=\cos(x)$
    $\displaystyle \Leftrightarrow \frac{3}{5\sqrt{50}} - \frac{4}{5}\left(\frac{7}{\sqrt{50}}\right)=\cos(x )$
    $\displaystyle \Leftrightarrow \frac{3}{25\sqrt{2}} - \frac{28}{25\sqrt{2}}=\cos(x)$
    $\displaystyle \Leftrightarrow \cos(x) = \frac{-\sqrt{2}}{2}$
    $\displaystyle \Leftrightarrow x = \frac{3\pi}{4}+2k\pi$
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    Re: arccos+arccot

    Quote Originally Posted by Soroban View Post
    Hello, Petrus!


    Let $\displaystyle \alpha \,=\,\arccos\left(\tfrac{3}{5}\right) \quad\Rightarrow\quad \cos\alpha \,=\,\tfrac{3}{5} \,=\,\tfrac{adj}{hyp}$
    . . Then: .$\displaystyle opp \,=\,4\quad\Rightarrow\quad \tan\alpha \,=\,\tfrac{4}{3}$

    Let $\displaystyle \beta \,=\,\text{arccot}\left(\tfrac{1}{7}\right) \quad\Rightarrow\quad \cot\beta \,=\,\tfrac{1}{7}$
    . . Then: .$\displaystyle \tan\beta \,=\,\tfrac{7}{1} \,=\,7$


    We have: .$\displaystyle \theta \;=\;\alpha + \beta$

    . . $\displaystyle \tan\theta \;=\;\frac{\tan\alpha + \tan\beta}{1 - (\tan\alpha)(\tan\beta)} \;=\;\frac{\frac{4}{3}+7}{1-\frac{4}{3}(7)} \;=\;\frac{\frac{25}{3}}{\text{-}\frac{25}{3}} \;=\;-1 $


    $\displaystyle \text{Therefore: }\:\theta \;=\;\frac{3\pi}{4} + \pi n\;\text{ for any integer }n.$

    Was it necessary to use $\displaystyle \tan\theta$? I tried using $\displaystyle \sin\theta$, but I ended up getting $\displaystyle \sin\frac{5}{\sqrt(50)}$
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  5. #5
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    Re: arccos+arccot

    Hello soroban!
    Idk where u got that formel tan o=tan a+tan b/... Etc
    Then i wanna ask how do i see what tan o=-1 ik its writen on My book but on test i cant use My book so plz can some1 tell me? Ik for 30degree,60 and 45 i can make a triangel but how i do with rest!!!
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