Question

1st step - label the sides of the triangle
angle 50: opposite side is 0.766, adjacent side is 0.643, hypotenuse is 1.909
angle 40: opposite side is 0.643, adjacent side is 0.766, hypotenuse is 1.909
step 2 - setup proportions / find ratios
angle 50: adjacent / hypotenuse = ratio @ 50° --> 3 : z = 0.643
angle 40: opposite / hypotenuse = ratio @ 40° --> 3 : z = 0.643

Explanation:

Step1: Use cosine for 50° angle

$\cos(50^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{3}{z}$

Step2: Solve for z

$z = \frac{3}{\cos(50^\circ)}$
We know $\cos(50^\circ) \approx 0.643$, so:
$z = \frac{3}{0.643} \approx 4.666$

Answer:

$z \approx 4.67$