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    Find all solutions for and exact value of...

    1) Find all solutions θ
    0 ≤ θ < 360 to the equation cos(2θ) = cosθ


    2) Find the exact value of cot(795)




    I have no idea how to do these, any help or tips?
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    Quote Originally Posted by alexcross View Post
    1) Find all solutions θ
    0 ≤ θ < 360 to the equation cos(2θ) = cosθ


    2) Find the exact value of cot(795)




    I have no idea how to do these, any help or tips?
    For 1. you have to use some Double Angle identities.

    \cos{(2\theta)} = \cos^2{\theta} - \sin^2{\theta} = \cos^2{\theta} - (1 - \cos^2{\theta}) = 2\cos^2{\theta} - 1.


    So if \cos{(2\theta)} = \cos{\theta}

    2\cos^2{\theta} - 1 = \cos{\theta}

    2\cos^2{\theta} - \cos{\theta} - 1 = 0

    This is a quadratic equation. Let X = \cos{\theta}

    2X^2 - X - 1 = 0

    2X^2 - 2X + X - 1 = 0

    2X(X - 1) + 1(X - 1) = 0

    (X - 1)(2X + 1) = 0


    So Case 1:

    X - 1 = 0 \implies \cos{\theta} - 1 = 0 \implies \cos{\theta} = 1

    Case 2:

    2X + 1 = 0 \implies 2\cos{\theta} +1 = 0 \implies \cos{\theta} = -\frac{1}{2}.


    Solve both cases for \theta over the interval 0 \leq \theta < 360^\circ.
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    Quote Originally Posted by alexcross View Post
    1) Find all solutions θ
    0 ≤ θ < 360 to the equation cos(2θ) = cosθ


    2) Find the exact value of cot(795)




    I have no idea how to do these, any help or tips?
    2. \cot{795^\circ} = \cot{(2\times 360^\circ + 75^\circ)} = \cot{75^\circ}

    \cot{75^\circ} = \frac{1}{\tan{75^\circ}}

     = \frac{1}{\tan{(45^\circ + 30^\circ)}}


    To evaluate this use the sum formula for tangent.

    \tan{(\alpha + \beta)} = \frac{\tan{\alpha} + \tan{\beta}}{1 - \tan{\alpha}\tan{\beta}}.

    Can you go from here?
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    Quote Originally Posted by alexcross View Post
    1) Find all solutions θ
    0 ≤ θ < 360 to the equation cos(2θ) = cosθ


    2) Find the exact value of cot(795)




    I have no idea how to do these, any help or tips?
    1. change \cos(2\theta) to 2\cos^2{\theta} - 1 ...

    2\cos^2{\theta} - 1 = \cos{\theta}

    2\cos^2{\theta} - \cos{\theta} - 1 = 0

    factor and solve for \theta


    2. 795 - 2(360) = 75 , a coterminal angle

    \cot(75) = \frac{\cos(45+30)}{\sin(45+30)}

    use your sum identities for cosine and sine, then evaluate.
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