cos-x-sin-x-1-2-cos-x-sin-x-3-8-pi-lt-x-lt-2pi-cos-x-sin-x-

Question Number 89193 by jagoll last updated on 16/Apr/20

\cos x - \sin x = \frac{1}{2}

\cos x \sin x = \frac{3}{8}, π < x < 2 π

\cos x + \sin x = ?

Commented by Tony Lin last updated on 16/Apr/20

∵ π < x < 2 π

∴ s i n x < 0

∵ c o s x s i n x = \frac{3}{8}

∴ c o s x < 0

{(c o s x + s i n x)}^{2} = {(c o s x - s i n x)}^{2} + 4 c o s x s i n x

= \frac{1}{4} + \frac{3}{2} = \frac{7}{4}

\Rightarrow c o s x + s i n x = - \frac{\sqrt{7}}{2}

{\begin{cases} c o s x + s i n x = - \frac{\sqrt{7}}{2} \\ c o s x - s i n x = \frac{1}{2} \end{cases}

\Rightarrow {\begin{cases} s i n x = \frac{- \sqrt{7} - 1}{4} \\ c o s x = \frac{- \sqrt{7} + 1}{4} \end{cases}

Commented by john santu last updated on 16/Apr/20

l e t \cos x + \sin x = p, p < 0

\frac{1}{2} p = \cos 2 x (i)

(i i) 2 \sin x \cos x = \frac{3}{4} \Rightarrow \sin 2 x = \frac{3}{4}

\cos 2 x = - \frac{\sqrt{7}}{4}

∴ \frac{1}{2} p = - \frac{\sqrt{7}}{4} \Rightarrow p = - \frac{\sqrt{7}}{2}

\cos x + \sin x = - \frac{\sqrt{7}}{2}

Commented by jagoll last updated on 16/Apr/20

t h a n k y o u

Facebook Tweet Pin