We're asked to express the angle
65 degrees 18 minutes 48 seconds only in degrees,
then express the angle in radians
as an exact fraction of pi.
So we are given the angle as 65 degrees,
18 minutes,
48 seconds,
the first step is to convert
18 minutes and 48 seconds to degrees,
well 60 minutes is equal to one degree
and therefore one minute is equal to one 60th of a degree
and because we have 18 minutes,
we can express 18 minutes
as 18 60ths of a degree,
so let's write this as 65 degrees
plus 18 60ths
of a degree,
similarly 3600 seconds is equal to one degree
and therefore one second is equal
to one 3600th of a degree
and because we have 48 seconds,
we can write 48 seconds
as 48 3600ths
of a degree, which gives us
plus 48
3600ths of a degree.
Now that we have everything in degrees,
let's simplify these two fractions.
To simplify 18 60ths,
we divide the numerator and denominator by six,
this simplifies to three 10ths,
to simplify 48 3600ths,
we divide the numerator and denominator by 48,
because 48 is the greatest common factor
between the numerator and denominator,
this simplifies to one over 75,
so now we can write the angle as 65 degrees
plus three 10ths of a degree
plus one 75th
of a degree
and now let's find the sum of the two fractions.
We have three 10ths
plus one 75th,
the least common denominator is 150
and therefore we multiply the numerator and denominator
of three 10ths by 15
and the numerator and denominator of one 75th by two.
Notice now we have a common denominator of 150
and the numerator is three times 15, which is 45
plus one times two, which is two
giving us 47 150ths
and therefore we can express the angle
as 65
and 47
150ths degrees,
so this is the exact angle expressed only using degrees.
Let's also express this using a decimal approximation.
This will be equal to 65.,
to determine the decimal, we divide 47 by 150,
we round to four decimal places,
47 divided by 150 is approximately .3133.
So the exact measure of the angle in degrees
is here as a mixed number
and here we have a decimal approximation in degrees.
Now for the second part, we want to express the angle
in radians as an exact fraction of pi.
We'll begin with the angle as a mixed number in degrees,
which is 65
and 47
150ths degrees
and now let's convert this to an improper fraction,
the denominator is going to remain 150
and the numerator is going to be 150
times 65
plus the numerator of 47
and the units here are still in degrees,
150 times 65
is equal to 9750
plus 47,
which gives us 9797,
so we can express the angle as an improper fraction
as 9797
150ths degrees,
we've now to convert from degrees to radians,
we multiply by pi
over 180 degrees.
An easy way to remember the 180 degrees
is in the denominator
is if we have degrees divided by degrees,
the units of degrees will simplify out,
leaving us with radians.
So now multiplying,
in the numerator, we have 9797 times pi,
in the denominator, we have 150 times 180,
which is equal to 27000
and this fraction does not simplify
and therefore this is the exact angle
expressed in radians in terms of pi.
We can also express this as
9797
27000ths
pi,
either form is acceptable.
I hope you found this helpful.