How many times are the hands of a clock at $90$ degrees.

Yes, but a more “mathematical” approach might be this: In a 12 hour period, the minute hand makes 12 revolutions while the hour hand makes one. If you switch to a rotating coordinate system in which the hour hand stands still, then the minute hand makes only 11 revolutions, and so it is at right angles with the hour hand 22 times. In a 24 hour day you get 2×22=44.

More mathematically, it can be done as :

The minute hand moves 360 degrees in 60 minutes. This means that the angle of the minute hand is given by 6t, where t is number of minutes past midnight.

The hour hand moves 30 degrees in 60 minutes. This means that the angle of the hours hand is given by 0.5t.

The hands start together at midnight. The first time they make a 90 degree angle is when the minute hand has moved 90 degrees further than the hour hand, so this is given by the equation:

6t = 0.5t + 90

5.5t = 90

t = 16 4/11 (16 minutes and 4/11 seconds)

In other words about 16 minutes past midnight.

The next time is when the minutes hand has gained another 180 degrees on the hour hand, and is 90 degrees behind it:

6t = 0.5t + 270

5.5t = 270

t = 49 1/11 (49 minutes and 1/11 seconds)

At about 11 minutes to 1 o'clock.

For every 180 degrees that the minute hand gains on the hour hand there will be one 90 degree angle, so every 49 1/11 - 16 4/11 = 32 8/11 minutes

24 hours is 1440 minutes. 1440/(32 8/11 ) = 44

So every 24 hours there are 44 right angles between minute hand and second hand.

Hope it helps :)

The question reads, "How many times are the hands of a clock at right angle in a day?"

There are three hands on a clock.
The number of occurrences of right angles between the hour and minute hands in a 24-hour period is 44.
The number of occurrences of right angles between the minute and second hands in a 24-hour period is 2832.
The number of occurrences of right angles between the second and hour hands in a 24-hour period is 2876.
Soooo-ooo: 5752.

To the down-voter(s): Is my answer wrong, or do you not like the typography, or what?