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Math Help - trigonometric functions

  1. #1
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    trigonometric functions

    Math trigonometric questions?

    1)Determine tan2θ if sinθ + cosθ = -1

    2) Express sin2θ in terms of csc θ and sec θ

    3) Express cos3θ in terms of sin θ
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  2. #2
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    Quote Originally Posted by crosser43 View Post
    Math trigonometric questions?

    1)Determine tan2θ if sinθ + cosθ = -1

    2) Express sin2θ in terms of csc θ and sec θ

    3) Express cos3θ in terms of sin θ
    1) sinθ + cosθ = -1
    sinθ + cosθ + 1 = 0
    2sin(0.5θ)cos(0.5θ) + 2cos^2(0.5θ) = 0
    2*cos(0.5θ)[sin(0.5θ) + cos(0.5θ)] = 0
    cos(0.5θ) = 0 or 0.5θ = 0.5π or θ = π
    sin(0.5θ) + cos(0.5θ) = 0
    tan(0.5θ) = -1 or 0.5θ = 0.25π or 2θ = π
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  3. #3
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    Trig

    Hello crosser43
    Quote Originally Posted by crosser43 View Post
    Math trigonometric questions?

    1)Determine tan2θ if sinθ + cosθ = -1
    Square both sides:

    (\sin\theta+\cos\theta)^2=1

    \Rightarrow \sin^2\theta +2\sin\theta\cos\theta+\cos^2\theta = 1

    Now use the identities \sin^2\theta +\cos^2\theta = 1 and 2\sin\theta\cos\theta = \sin2\theta:

    \Rightarrow 1 + \sin2\theta = 1

    \Rightarrow \sin2\theta = 0

    \Rightarrow \tan2\theta = \frac{\sin2\theta}{\cos2\theta}=0

    2) Express sin2θ in terms of csc θ and sec θ
    \sin2\theta= 2\sin\theta\cos\theta = 2\left(\frac{1}{\csc\theta}\right)\left(\frac{1}{\  sec\theta}\right)=\frac{2}{\csc\theta\sec\theta}

    3) Express cos3θ in terms of sin θ
    See line 40 of Multiple-Angle Formulas -- from Wolfram MathWorld, and put \cos\theta = \sqrt{1-\sin^2\theta}

    Grandad
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