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Math Help - Trigonometry

  1. #1
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    Trigonometry

    can u help with these 2 exercises?: 1) which are the solutions to the equation tanx=2cotx,in the interval -180<x<180?...2) Which is the value of arcsin1/2-arcsin(-1/2).. i will be glad if you could help me
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  2. #2
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    Re: Trigonometry

    \displaystyle \begin{align*} \tan{(x)} &= 2\cot{(x)} \\ \frac{\sin{(x)}}{\cos{(x)}} &= \frac{2\cos{(x)}}{\sin{(x)}} \\ \sin^2{(x)} &= 2\cos^2{(x)} \\ 1 - \cos^2{(x)} &= 2\cos^2{(x)} \\ 1 &= 3\cos^2{(x)} \\ \frac{1}{3} &= \cos^2{(x)} \\ \pm \frac{1}{\sqrt{3}} &= \cos{(x)} \\ x &= \left\{ \arccos{\left( \frac{1}{\sqrt{3}} \right)}, \pi - \arccos{\left( \frac{1}{\sqrt{3}} \right)}, \pi + \arccos{\left( \frac{1}{\sqrt{3}} \right)} , 2\pi - \arccos{\left( \frac{1}{\sqrt{3}} \right)} \right\} + 2\pi \, n \textrm{ where } n \in \mathbf{Z} \\ x &= \left\{ \arccos{\left( \frac{1}{\sqrt{3}} \right)} - \pi , -\arccos{\left( \frac{1}{\sqrt{3}} \right)} , \arccos{\left( \frac{1}{\sqrt{3}} \right)} , \pi - \arccos{\left( \frac{1}{\sqrt{3}} \right)} \right\} \textrm{ in the interval specified}  \end{align*}

    Edit: I naturally put the answer in radians when it should have been degrees. You can fix that
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  3. #3
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    Re: Trigonometry

    Hello, endri!

    \text{1) What are the solutions to the equation: }\,\tan x\,=\,2\cot x,\,\text{ on the interval }(\text{-}180^o,180^o)

    We have: . \tan x \:=\:\frac{2}{\tan x} \quad\Rightarrow\quad \tan^2x \:=\:2 \quad\Rightarrow\quad \tan x  \:=\:\pm\sqrt{2}

    Therefore: . x \;\approx\;\{\text{-}125.3^o,\:\text{-}54.7^o,\:54.7^o,\:125.3^o\}




    \text{2) What is the value of: }\,\arcsin(\tfrac{1}{2})-\arcsin(\text{-}\tfrac{1}{2})

    Using principal values: . \begin{Bmatrix}\arcsin(\frac{1}{2}) &=& \frac{\pi}{6} \\ \\[-4mm] \arcsin(\text{-}\frac{1}{2}) &=& \text{-}\frac{\pi}{6} \end{Bmatrix}

    Therefore: . \arcsin(\tfrac{1}{2}) - \arcsin(\text{-}\tfrac{1}{2}) \;=\;\frac{\pi}{6} - \left(\text{-}\frac{\pi}{6}\right) \;=\;\frac{\pi}{3}
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  4. #4
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    Re: Trigonometry

    thanks for your help man
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