Question

If k ≥ 1, the graphs of y = sinx and y = keex
intersect for
x ≥ 0. Find the smallest value of k for which the graphs are tan￾gent.
k =
What are the coordinates of the point of tangency?
x = , y = .

Answer 1

For tangency,

$sin x=ke^{-x}$ and $cos x= -ke^{-x}$

$sinx=-cos x=ke^{-x}$

$tanx=-1 for x= nπ+3π/4$

$1/\sqrt{2}=ke^{-3π/4} ; x = 3π/4 ; n=0$

$k=\frac {e^{3π/4}}{\sqrt{2}}$

Coordinates of the point of tangency $(\frac{3π}{4},\frac 1{\sqrt{2}})$