Question

How to convert 10 radian in degree minutes and seconds ?

Answer 1

`572^@ 57'36''`

Explanation:

1. converting to degree : ( Multiply by `180/pi` )
   `10xx180/pi=572. color(red)(96)^@`

2. converting to minutes :( `color(red)(0.96) xx 60` )
   `color(white)(.)`
   `0.96xx60=57. color(blue)(6)`

3. converting into seconds : ( `color(blue)(0.6) xx 60` )
   `color(white)(.)`
   `0.6xx60=36`

So , `10` radians `=572^@ 57'36''`

Answer 2

10 rad `~~ 572^@57'28`”

Explanation:

We need to know what a radian is to answer this. An angle with an arc that has the same length as the radius of a circle is called a radian. As the circumference of a circle with radius = 1 har a length of `2pi`, it follows that `π` rad = `180°`

Therefore 10 rad = `(180^@*10)/pi = 572.95779513^@`

As there are `60'` in `1^@` and `60`” in `1'`, we get
`0.95779513^@=0.95779513*3600`”`~~3448`”`=57'28`”

Therefore we get:
10 rad `~~ 572^@57'28`”

Please note that this is more than one time around the circle, by the way.

And by the way, please note that the discrepancy between my and @Evan's answer is due to the level of accuracy used.

Answer 3

`572^o color(white)("..")57^("'") color(white)("..")28^("''")` to the nearest second

Explanation:

`color(red)("Important fact: there are "2pi" radians in a circle")`

So `ul("1 radian")` in degrees is `360/(2pi) = 180/pi" degrees"`

We have 10 radians so the count of degrees is

`10xx180/pi = 1800/pi" Degrees" ~~ 572.957795....`

Store this in your calculator memory to reduce rounding errors.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So the degrees part of the measure is: `572^o`

The minutes part is `(1800/pi-572)xx60 = 57.467..=57^'`

The second's part is:
`(1800/pi-572-57/60)=7.79513...xx10^-3" degrees"`

Convert this into seconds: `7.79513...xx10^-3-:1/(60^2)=28.0624.... "seconds"`

So the answer is: `572^o color(white)("..")57^("'") color(white)("..")28^("''")` to the nearest second