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Thread: weird trig question?

  1. #1
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    weird trig question?

    Determine 'a' if

    $\displaystyle \cos(\theta+{\frac{\pi}{4}}) = a(\cos\theta + \sin\theta)$


    Using the identity:

    $\displaystyle \cos(\theta+\phi)$ $=$ $\cos\theta\cdot\cos\phi-\sin\theta\cdot\sin\phi$

    I have got
    $\displaystyle \begin{eqnarray}
    {\dfrac{1}{\sqrt{2}}}\left(\cos\theta-\sin\theta\right) &=&a\left(\cos\theta + \sin\theta\right)\nonumber
    \end{eqnarray}
    $

    But don't know where to go from here??
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  2. #2
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    Are you sure it wasn't asking you to find $\displaystyle \displaystyle a$ if

    $\displaystyle \displaystyle \cos{\left(\theta - \frac{\pi}{4}\right)} = a(\cos{\theta} + \sin{\theta})$?
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  3. #3
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by mathswannabe View Post
    Determine 'a' if

    $\displaystyle \cos(\theta+{\frac{\pi}{4}}) = a(\cos\theta + \sin\theta)$


    Using the identity:

    $\displaystyle \cos(\theta+\phi)$ $=$ $\cos\theta\cdot\cos\phi-\sin\theta\cdot\sin\phi$

    I have got
    $\displaystyle \begin{eqnarray}
    {\dfrac{1}{\sqrt{2}}}\left(\cos\theta-\sin\theta\right) &=&a\left(\cos\theta + \sin\theta\right)\nonumber
    \end{eqnarray}
    $

    But don't know where to go from here??
    $\displaystyle \dfrac{1}{\sqrt{2}}(\cos(\theta)-\sin(\theta))=a(\cos(\theta)+\sin(\theta))$

    $\displaystyle a = \dfrac{1}{\sqrt{2}} \cdot \dfrac{(\cos(\theta)-\sin(\theta))}{(\cos(\theta)+\sin(\theta))}$

    you can simplify further as:

    $\displaystyle a = \dfrac{1}{\sqrt{2}} \cdot \dfrac{(\cos(\theta)-\sin(\theta))}{(\cos(\theta)+\sin(\theta))} \times \dfrac{(\cos(\theta)+\sin(\theta))}{(\cos(\theta)+ \sin(\theta))}=...$
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