Question

Convert `5^@ 37^'30^('')`into radians?

Answer 1

`673/21600 pi " radians"larr color(red)("Exact answer")`

`color(white)("d")`

`x=0.0979" radians" color(red)(larr" Approx answer")`
`color(white)("dddddddddddddddddd.d")`rounded to 4 dec places

Explanation:

Measuring in degrees `37' = 36/60 = 3/5` degrees

and `30''=cancel(30)^1/(cancel(60)^2xx60)=1/(120` degrees

Therefor `5^ocolor(white)(.)37'color(white)(.)30''=5+3/5+1/120=5 73/120 = 673/120` degrees

Used fractions because the decimal equivalent has a repeating value. If I use decimals I may introduced a rounding error. Also `pi` is an irrational number so converting to decimal will have a rounding error.
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Known: `1/2` a circle has `pi" radians "=180^o`

When conducting conversions a good way of thinking about it is to use ratio but in fraction format.

Let the unknown count of radians be `x` giving

`("required radians")/("radians in "1/2" a circle") -=( 673/120)/(180)`

`color(white)("dddddddd")(x)/(pi)color(white)("dddddddd") -=( 673/120)/(180)`

Multiply both sides by `pi`

`x=( 673/120)/180xxpi`

`x= 673/120xx1/180xxpi`

`x=673/21600 pi larr color(red)("Exact answer")`

`x=0.097883882....`

`x=0.0979" radians" color(red)(larr" Approximate answer")`