Cho Sin α + cos α = ¾ Tính P= sin α( cos α +15)

Question

Cho Sin α + cos α = ¾
Tính P= sin α( cos α +15)

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Peyton 3 tuần 2021-11-19T20:23:45+00:00 2 Answers 2 views 0

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    0
    2021-11-19T20:24:56+00:00

    $\sin a+\cos a=\dfrac{3}{4}$     (1)

    $\Rightarrow 1+2\sin a+\cos a=dfrac{9}{16}$

    $\Leftrightarrow \sin a.\cos a=\dfrac{-7}{32}$      (2)

    (1)(2)$\Rightarrow (\sin a;\cos a)=(\pm \dfrac{3+\sqrt{23}}{8}; \pm \dfrac{3-\sqrt{23}}{8})=(\pm \dfrac{3-\sqrt{23}}{8};\dfrac{3+\sqrt{23}}{8})$

    (thay lần lượt 4 cặp $(\sin a;\cos a)$ tìm P.

    0
    2021-11-19T20:25:25+00:00

    Giải thích các bước giải:

     Ta có:

    \(\begin{array}{l}
    \sin a + \cos a = \frac{3}{4} \Rightarrow \cos a = \frac{3}{4} – \sin a\\
    {\sin ^2}a + {\cos ^2}a = 1\\
     \Leftrightarrow {\sin ^2}a + {\left( {\frac{3}{4} – \sin a} \right)^2} = 1\\
     \Leftrightarrow {\sin ^2}a + \frac{9}{{16}} – \frac{3}{2}\sin a + {\sin ^2}a = 1\\
     \Leftrightarrow 2{\sin ^2}a – \frac{3}{2}\sin a – \frac{7}{{16}} = 0\\
     \Leftrightarrow \sin a = \frac{{3 \pm \sqrt {23} }}{8}\\
    \sin a + \cos a = \frac{3}{4}\\
     \Leftrightarrow {\sin ^2}a + 2\sin a.\cos a + {\cos ^2}a = \frac{9}{{16}}\\
     \Leftrightarrow 1 + 2\sin a.\cos a = \frac{9}{{16}}\\
     \Rightarrow \sin a.\cos a =  – \frac{7}{{32}}\\
     \Rightarrow P = \sin a.\cos a + 15\sin a =  – \frac{7}{{32}} + 15.\frac{{3 \pm \sqrt {23} }}{8} = ….
    \end{array}\)

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