The sexagesimal numeration system

Lucía is practicing with her telescope and has turned the lens through an angle of $120^\circ$. If she then decides to reverse the turn by $35^\circ$ $15'$, what angle does her telescope show now?

In an esports championship, the first game of a final lasted $45$ min and $38$ s. After a short break, the second game lasted $52$ min and $45$ s. Calculate the total duration of both games combined and express the result in an appropriate time format.

A professional diving team has an air supply that allows them to stay underwater for $3$ hours and $15$ minutes. If they must divide this time equally to explore $5$ different sectors of a reef, how much time should they spend observing each sector?

In a technical drawing class, we are asked to draw the complementary angle of a given angle. If the original angle measures $34^\circ 52' 18''$, carry out the necessary calculations to find the exact measure of the angle needed to complete the right angle ($90^\circ$).

A lighthouse emits three different light signals to guide ships:

• The red light flashes every $18$ seconds.

• The green light flashes every $45$ seconds.

• The white light flashes every $1$ minute and $15$ seconds.

If at $22:00:00$ at night all three lights flash together for the first time:

1. a) How much time will pass until they flash together again? Express the result in minutes and seconds.
2. b) At what exact time will they flash together for the second time?
3. c) How many times will all three lights flash together from $22:00:00$ to $00:00:00$, both times included?

A video editor is working on a short film for social media. The project has a strict time limit of exactly $10$ minutes. They have four clips with the following durations:

• Clip 1: $2$ min $14$ s $45$ hundredths.

• Clip 2: $155$ s $80$ hundredths.

• Clip 3: $3$ min $12$ s.

• Clip 4: $1$ min $58$ s $95$ hundredths.

Calculate the exact time left over or needed to reach the limit of $10$ minutes. (Remember that $1$ second has $100$ hundredths).

In an irregular hexagon, the measurements of five of its interior angles are known:

• $A = 110^\circ \ 15' \ 20''$

• $B = 125^\circ \ 40'$

• $C = 98^\circ \ 25''$

• $D = 130^\circ \ 12' \ 15''$

• $E = 115^\circ \ 50' \ 30''$

Given that the sum of the interior angles of a $n$-sided polygon is $S = (n - 2) \cdot 180^\circ$, calculate the measure of the sixth angle $F$ in compound form.

A surveillance drone carries out a security sweep by rotating on its axis. It starts at a position of $0^\circ$ and makes the following consecutive movements:

1. Turns to the right through an angle of $125^\circ \ 45' \ 52''$.

2. Turns to the left by $87^\circ \ 50' \ 15''$.

3. Makes a third turn to the right that is exactly three times the angle resulting from subtracting the previous two movements.

Calculate the final position of the drone relative to its starting point in degrees, minutes and seconds.