Calculus and Sin function

Question

Find all the quatrains of points $ (x_1,f(x_1)),(x_2,f(x_2)),(x_3,f(x_3)),(x_4,f(x_4))$ on the graph of $f(x)=\sin x$ such that the tangents by the four points form a rectangle.

Could you help me to solve this out?

Answer 1

We need that

• $f'(x_i)\cdot f'(x_j)=-1\implies \cos x_i\cdot \cos x_j=-1\implies x_i=0\quad x_j =\pi$

thus we can choose

• $x_1=0$
• $x_2=\pi$
• $x_3=2\pi$
• $x_4=3\pi$