Question

Find the equation of tangents to the curve $y=\cos x$, that are parallel to the line $x+2y=0$

Answer 1

$y=\cos x$

The slope of equation is given as $\frac{dy}{dx}=\frac{d}{dx}\cos x\frac{dy}{dx}=\sin x$

The slope of tangent parallel to $x+2y=0$ is $\frac{-1}{2}$

$⟹\sin x=\frac{-1}{2}\sin x=\sin -\frac{\pi }{6}x=-\frac{\pi }{6}$

$⟹y=\cos -\left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$

The equation of tangent is

$y-\frac{\sqrt{3}}{2}=\frac{-1}{2}(x+\frac{\pi }{6})2y-\sqrt{3}+x+\frac{\pi }{6}=06x+12y-6\sqrt{3}+\pi =0$