Question

1) two hot-air balloons are 48.2 m and 61.0 m above the ground. A person in the left balloon observes that the right balloon is 13.3° above the horizontal. What is the horizontal distance x between the two balloons?

The left balloon is 48.2m and the right balloon is 61.0m.

I am not sure how to start this problem.

Answer 1

tan13.3º =(61-48.2)/x
x =12.8/tan13.3º =54.1 m

Answer 2

How do you know when to use tan instead of cos or sin?

Answer 3

Horizontal leg (the first cathetus) is the distance between vertical lines connecting each of the baloons with the ground; the vertical leg (the second cathetus) is difference between heights; the hypothenuse is separation between two baloons.
Accotding to difinition
tan α = opposite side/adjacent side.
In the problem you know the opposite side (the second cathetus) and the angle, and you must find the first cathetus.

Answer 4

Horizontal leg (the first cathetus) is the distance between vertical lines connecting each of the baloons with the ground; the vertical leg (the second cathetus) is difference between heights; the hypothenuse is separation between two baloons.
Accotding to difinition
tan α = opposite side/adjacent side.
In the problem you know the opposite side (the second cathetus) and the angle, and you must find the first cathetus.