Question

How do I demonstrate that there is only one value of x (between 0 and 2pi) that verifies cosx=3/5 and sinx=4/5

Answer 1

well, as we can see, the cos(x) as well as the sin(x) are both positive, that puts the angle in the I Quadrant only, everywhere else they differ in signs or are both negative.

$\bf cos(x)=\cfrac{\stackrel{adjacent}{3}}{\stackrel{hypotenuse}{5}}\qquad \qquad \qquad sin(x)=\cfrac{\stackrel{opposite}{4}}{\stackrel{hypotenuse}{5}}$

that simply means it has an adjacent side of 3, opposite side of 4, and a hypotenuse of 5, like you see it in the picture below.

and we can get the inverse function of either function to get about 53.130102354° for ∡x.