Multiple Choice Question (MCQ)

If α and β are angles in the first quadrant \tanα=\dfrac{1}{7}, \sinβ=\dfrac{1}{\sqrt{10} } , then the value of α+2β is

  1. ×

    0\degree

  2. ×

    30\degree

  3. 45\degree

  4. ×

    60\degree

Collected in the board: Trigonometry

Steven Zheng posted 4 hours ago

Answer

  1. Using double angle identity

    \cos 2β = 1-2 \sin^2β

    = 1-2\Big( \dfrac{1}{\sqrt{10} } \Big) ^2

    =\dfrac{4}{5}

    Then

    \sin 2β =\dfrac{3}{5}

    \begin{aligned} \tanα=\dfrac{1}{7} \implies &\sinα = \dfrac{1}{\sqrt{1+7^2} } = \dfrac{1}{\sqrt{50} } = \dfrac{1}{5\sqrt{2} } \\ &\cos α = \dfrac{7}{5\sqrt{2} } \end{aligned}

    \cos( α +2β ) = \cos α\cos 2β -\sin α \sin 2β

    = \dfrac{7}{5\sqrt{2} }\cdot \dfrac{4}{5} - \dfrac{1}{5\sqrt{2} }\cdot \dfrac{3}{5}

    =\dfrac{\sqrt{2} }{2}

    Therefore

    α +2β = 45\degree

    C is the correct choice

Steven Zheng posted 3 hours ago

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