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

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

2. Giải thích các bước giải:

Ta có:

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