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Math Help - Find the Value

  1. #1
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    Find the Value

    Let t = theta for short

    (1) If cos(t) = 0.2, find the value of:
    cos(t) + cos(t + 2pi) + cos(t + 4pi)

    (2) If cot(t) = -2, find the value of:
    cot(t) + cot(t - pi) + cot(t - 2pi)
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  2. #2
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    Hi magentarita,

    Note that \cos (x)~\equiv~\cos (x+2k\pi)~\forall k \in \mathbb{Z} and that \cos (x-\pi)~\equiv~-\cos (x)
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  3. #3
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    Hello, magentarita!

    Another approach . . .


    (1) If \cos\theta = 0.2, find the value of: . \cos\theta + \cos(\theta + 2\pi) + \cos(\theta + 4\pi)
    Identity: . \cos(A + B) \;=\;\cos(A)\cos(B) - \sin(A)\sin(B)


    We have: . \cos\theta + \cos(\theta + 2\pi) + \cos(\theta+4\pi)

    . . . . . . = \;\cos\theta + \bigg[\cos\theta\cos2\pi-\sin\theta\sin2\pi\bigg] + \bigg[\cos\theta\cos4\pi - \sin\theta\sin2\pi\bigg]

    . . . . . . = \;\cos\theta + \bigg[\cos\theta\cdot1 - \sin\theta\cdot0\bigg] + \bigg[\cos\theta\cdot1 - \sin\theta\cdot0\bigg]


    . . . . . . = \;3\cos\theta \;=\;3(0.2) \;=\;\boxed{0.6}




    (2) If \cot\theta = -2, find the value of: . \cot\theta + \cot(\theta - \pi) + \cot(\theta - 2\pi)
    Identity: . \cot(A - B) \;=\;\frac{1}{\tan(A - B)} \;=\;\frac{1 +\tan(A)\tan(B)}{\tan(A) - \tan(B)}


    We have: . \cot\theta + \cot(\theta-\pi) + \cot(\theta-2\pi)

    . . . . . . =\;\cot\theta + \bigg[\frac{1 + \tan\theta\tan\pi}{\tan\theta - \tan\pi}\bigg] + \bigg[\frac{1 + \tan\theta\tan2\pi}{\tan\theta - \tan2\pi}\bigg]

    . . . . . . = \;\cot\theta + \bigg[\frac{1 + \tan\theta\cdot0}{\tan\theta - 0}\bigg] + \bigg[\frac{1 + \tan\theta\cdot0}{\tan\theta - 0}\bigg]

    . . . . . . = \;\cot\theta + \frac{1}{\tan\theta} + \frac{1}{\tan\theta} \;=\;\cot\theta + \cot\theta + \cot\theta


    . . . . . . = \;3\cot\theta \;=\;3(-2) \;=\;\boxed{-6}

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  4. #4
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    Ooops.... I misread cot for cos on question 2, thank you Soroban.
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  5. #5
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    Hey...

    I thank both of you, especially Soroban.
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