Answer:
To express the angle 380° 17' 23" in radians, we need to convert each part of the angle to radians and then sum them up.
1 degree (°) = π/180 radians
1 minute (') = 1/60 degrees
1 second (") = 1/3600 degrees
First, let's convert 380° to radians:
380° * (π/180) = 19π/9 radians
Next, let's convert 17 minutes to degrees:
17' * (1/60) = 17/60 degrees
Now, let's convert 23 seconds to degrees:
23" * (1/3600) = 23/3600 degrees
Now, let's add all the parts together:
19π/9 radians + 17/60 degrees + 23/3600 degrees
To add fractions, we need to find a common denominator, which in this case is 3600:
= (19π/9) + (17/60)*(60/60) + (23/3600)*(60/60)
= 19π/9 + 17/3600 + 23/129600
Now, we can add the fractions:
= (19π/9) + (6120/3600) + (23/129600)
= (19π/9) + (17/10) + (23/129600)
Now, let's convert 17/10 to an improper fraction:
= (19π/9) + (170/100) + (23/129600)
= (19π/9) + (17/10) + (23/129600)
Now, let's find a common denominator:
= (19π/9) + (22170/12960) + (23/129600)
Now, we can add the fractions:
= (19π/9) + (22170/12960) + (23/129600)
Now, let's simplify:
= (19π/9) + (3695/216) + (23/129600)
Now, let's add the fractions:
= (19π/9) + (3695/216) + (23/129600)
= (19π/9) + (3695/216) + (23/129600)
Therefore, the angle 380° 17' 23" is approximately equal to:
(19π/9) + (3695/216) + (23/129600) radians