Cho Sin α + cos α = ¾ Tính P= sin α( cos α +15) 19/11/2021 Bởi Peyton Cho Sin α + cos α = ¾ Tính P= sin α( cos α +15)
$\sin a+\cos a=\dfrac{3}{4}$ (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. Bình luận
Giải thích các bước giải: Ta có: \(\begin{array}{l}\sin a + \cos a = \frac{3}{4} \Rightarrow \cos a = \frac{3}{4} – \sin a\\{\sin ^2}a + {\cos ^2}a = 1\\ \Leftrightarrow {\sin ^2}a + {\left( {\frac{3}{4} – \sin a} \right)^2} = 1\\ \Leftrightarrow {\sin ^2}a + \frac{9}{{16}} – \frac{3}{2}\sin a + {\sin ^2}a = 1\\ \Leftrightarrow 2{\sin ^2}a – \frac{3}{2}\sin a – \frac{7}{{16}} = 0\\ \Leftrightarrow \sin a = \frac{{3 \pm \sqrt {23} }}{8}\\\sin a + \cos a = \frac{3}{4}\\ \Leftrightarrow {\sin ^2}a + 2\sin a.\cos a + {\cos ^2}a = \frac{9}{{16}}\\ \Leftrightarrow 1 + 2\sin a.\cos a = \frac{9}{{16}}\\ \Rightarrow \sin a.\cos a = – \frac{7}{{32}}\\ \Rightarrow P = \sin a.\cos a + 15\sin a = – \frac{7}{{32}} + 15.\frac{{3 \pm \sqrt {23} }}{8} = ….\end{array}\) Bình luận
$\sin a+\cos a=\dfrac{3}{4}$ (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.
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\sin a + \cos a = \frac{3}{4} \Rightarrow \cos a = \frac{3}{4} – \sin a\\
{\sin ^2}a + {\cos ^2}a = 1\\
\Leftrightarrow {\sin ^2}a + {\left( {\frac{3}{4} – \sin a} \right)^2} = 1\\
\Leftrightarrow {\sin ^2}a + \frac{9}{{16}} – \frac{3}{2}\sin a + {\sin ^2}a = 1\\
\Leftrightarrow 2{\sin ^2}a – \frac{3}{2}\sin a – \frac{7}{{16}} = 0\\
\Leftrightarrow \sin a = \frac{{3 \pm \sqrt {23} }}{8}\\
\sin a + \cos a = \frac{3}{4}\\
\Leftrightarrow {\sin ^2}a + 2\sin a.\cos a + {\cos ^2}a = \frac{9}{{16}}\\
\Leftrightarrow 1 + 2\sin a.\cos a = \frac{9}{{16}}\\
\Rightarrow \sin a.\cos a = – \frac{7}{{32}}\\
\Rightarrow P = \sin a.\cos a + 15\sin a = – \frac{7}{{32}} + 15.\frac{{3 \pm \sqrt {23} }}{8} = ….
\end{array}\)