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Thread: Difficult Trigo Problems

  1. #1
    Junior Member
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    Difficult Trigo Problems

    Hello people,
    Please help me with the following problems. I was unable to solve any of them.

    1) If $\displaystyle Cos(a) + Cos(b) + Cos(c) = Sin(a) + Sin(b) + Sin(c) = 0$,
    then the value of $\displaystyle Cos(3a) + Cos(3b) + Cos(3c)$ = ?

    2) If [e^cos(x)] - [e^-cos(x)] = 4. Find the value of $\displaystyle Cos(x)$

    3) If $\displaystyle 2Sin^2 \theta$ = $\displaystyle 3 Cos \theta$, find the positive angle of $\displaystyle \theta$

    4) In $\displaystyle \triangle ABC$, $\displaystyle \theta$ is an acute angle & $\displaystyle tan\theta$ is equal to three times of $\displaystyle Cot\theta$. Find the value of $\displaystyle (sin^2\theta) + (Cosec^2\theta) - (1/2) (Cot^2\theta)$
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  2. #2
    MHF Contributor Amer's Avatar
    Joined
    May 2009
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    Jordan
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    Quote Originally Posted by saberteeth View Post
    Hello people,
    Please help me with the following problems. I was unable to solve any of them.

    1) If $\displaystyle Cos(a) + Cos(b) + Cos(c) = Sin(a) + Sin(b) + Sin(c) = 0$,
    then the value of $\displaystyle Cos(3a) + Cos(3b) + Cos(3c)$ = ?

    2) If [e^cos(x)] - [e^-cos(x)] = 4. Find the value of $\displaystyle Cos(x)$

    3) If $\displaystyle 2Sin^2 \theta$ = $\displaystyle 3 Cos \theta$, find the positive angle of $\displaystyle \theta$

    4) In $\displaystyle \triangle ABC$, $\displaystyle \theta$ is an acute angle & $\displaystyle tan\theta$ is equal to three times of $\displaystyle Cot\theta$. Find the value of $\displaystyle (sin^2\theta) + (Cosec^2\theta) - (1/2) (Cot^2\theta)$

    $\displaystyle 2) e^{\cos x} - e^{-\cos x} = 4 $

    $\displaystyle e^{2\cos x } - 1 = 4 e^{\cos x} $

    $\displaystyle e^{2\cos x } - 4e^{\cos x} - 1 =0$

    let $\displaystyle u=e^{\cos x} $

    $\displaystyle u^2 - 4u - 1=0 $

    $\displaystyle u=\frac{4 \mp \sqrt{16+4}}{2}$

    $\displaystyle u=2+\sqrt{5} $ or $\displaystyle u =2-\sqrt{5}$

    $\displaystyle e^{\cos x} = 2+\sqrt{5} \Rightarrow \cos x = \ln (2+\sqrt{5} )$

    $\displaystyle e^{\cos x } = 2 -\sqrt{5} \Rightarrow \cos x = \ln (2-\sqrt{5})$


    3)$\displaystyle 2\sin ^2 \theta = 3\cos \theta $

    $\displaystyle 2(1-\cos ^2\theta) = 3\cos \theta $

    $\displaystyle 2\cos ^2\theta +3\cos \theta -2=0 $

    let $\displaystyle u=\cos \theta $

    $\displaystyle 2u^2+3u-2 =0 $ you can solve it after you find u values find x
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  3. #3
    MHF Contributor Amer's Avatar
    Joined
    May 2009
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    Jordan
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    1,093
    Quote Originally Posted by saberteeth View Post
    Hello people,
    Please help me with the following problems. I was unable to solve any of them.

    1) If $\displaystyle Cos(a) + Cos(b) + Cos(c) = Sin(a) + Sin(b) + Sin(c) = 0$,
    then the value of $\displaystyle Cos(3a) + Cos(3b) + Cos(3c)$ = ?

    2) If [e^cos(x)] - [e^-cos(x)] = 4. Find the value of $\displaystyle Cos(x)$

    3) If $\displaystyle 2Sin^2 \theta$ = $\displaystyle 3 Cos \theta$, find the positive angle of $\displaystyle \theta$

    4) In $\displaystyle \triangle ABC$, $\displaystyle \theta$ is an acute angle & $\displaystyle tan\theta$ is equal to three times of $\displaystyle Cot\theta$. Find the value of $\displaystyle (sin^2\theta) + (Cosec^2\theta) - (1/2) (Cot^2\theta)$

    $\displaystyle 4) \tan \theta = 3 \cos \theta $

    $\displaystyle \tan \theta = \frac{3}{\tan \theta }$

    $\displaystyle \tan ^2\theta = 3 $

    $\displaystyle \tan \theta = \sqrt{3} $ $\displaystyle \Rightarrow \theta =\frac{\pi}{3}$

    the rest of question just you should know

    $\displaystyle \sin \frac{\pi}{3} = ?? $

    $\displaystyle \sec \frac{\pi}{3}= ?? $

    $\displaystyle \cos \frac{\pi}{3} =??$
    the rest for you
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