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Math Help - Find sin(theta + pi/4)

  1. #1
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    Find sin(theta + pi/4)

    given that π/2 ≤ θ ≤ 3π/2 and sin θ = -2/7 find sin(θ+π/4)
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  2. #2
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    Re: Find sin(theta + pi/4)

    Hello, absolutjessy!

    \text{Given: }\:\tfrac{\pi}{2} \,\le\,\theta\,\le\, \tfrac{3\pi}{2}\,\text{ and }\,\sin\theta \,=\,\text{-}\frac{2}{7}
    \text{Find: }\:\sin\left(\theta + \tfrac{\pi}{4}\right)

    We are told that \theta is in Quad 2 or Quad 3.
    Sine is negative in Quad 3 and Quad 4.
    . . Hence, \theta is in Quad 3.

    \sin\theta \,=\,\frac{\text{-}2}{7} \,=\,\frac{opp}{hyp}

    \theta is in a right triangle with: opp = \text{-}2,\;hyp = 7
    . . Then: . adj \,=\, \text{-}\sqrt{45} \,=\,\text{-}3\sqrt{5} \quad\Rightarrow\quad \cos\theta \:=\:\text{-}\frac{3\sqrt{5}}{7}


    We have: . \sin\left(\theta + \tfrac{\pi}{4}\right) \;=\; \sin\theta\cos\tfrac{\pi}{4} + \cos\theta\sin\tfrac{\pi}{4}

    n . . . . . . . . . . . . . . . =\;\left(\text{-}\tfrac{2}{7}\right)\left(\tfrac{1}{\sqrt{2}} \right) + \left(\text{-}\tfrac{3\sqrt{5}}{7}\right)\left(\tfrac{1}{\sqrt{  2}}\right)

    n . . . . . . . . . . . . . . . =\;\frac{\text{-}2}{7\sqrt{2}} + \frac{\text{-}3\sqrt{5}}{7\sqrt{2}}

    n . . . . . . . . . . . . . . . =\;-\frac{2 + 3\sqrt{5}}{7\sqrt{2}}
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