homework and exercises - Which is the correct triangle to draw to derive the force component along a rope?

When solving for the force component ($F$) in the direction of the string, the following triangle is drawn.

(Adapted from Source)

Giving $F=mg * cos θ$

Why is it drawn that way, and not like this?

Which would instead give:

$F=mg / cos θ$

$ cos θ=mg/F$

$F=mg / cos θ$

Answer

For the sake of this explanation, I shall refer to the main force as opposite tension.

In these circumstances, the opposite tension force should never exceed the total mass force. You can consider extremes and seeing how these would affect the overall tension in the string. The extremes I will be using are when θ is equal to 0° and 90°.

0° Scenario:

Opposite tension force is equal to the mass force as it is the only component vector. mg/cos(0) = mg.

90° Scenario:

Opposite tension force approaches infinity. mg/cos(90) = ∞.

Since both extremes give logical conclusions when using mg cos(θ) as opposed to impossibilities, it is safe to assume that you should be treating the mass force as the hypotenuse rather than the opposite tension force.