To convert the angle [tex]\( \alpha = 53^\circ 47' 3'' \)[/tex] to decimal degrees, we'll follow these steps:
1. Understand the Components of the Angle:
- Degrees: [tex]\( 53^\circ \)[/tex]
- Minutes: [tex]\( 47' \)[/tex]
- Seconds: [tex]\( 3'' \)[/tex]
2. Convert Minutes to Decimal Degrees:
- There are 60 minutes in a degree. To convert the 47 minutes to decimal degrees, we divide by 60:
[tex]\[
\frac{47}{60} \approx 0.7833
\][/tex]
3. Convert Seconds to Decimal Degrees:
- There are 3600 seconds in a degree (since [tex]\( 1^\circ = 60' \times 60'' = 3600'' \)[/tex]). To convert the 3 seconds to decimal degrees, we divide by 3600:
[tex]\[
\frac{3}{3600} \approx 0.0008
\][/tex]
4. Summing Up the Contributions:
- The total angle in decimal degrees is given by the sum of the degrees, the contribution from the minutes, and the contribution from the seconds:
[tex]\[
53 + 0.7833 + 0.0008 = 53.7842
\][/tex]
5. Final Answer:
The angle [tex]\( \alpha \)[/tex] in decimal degrees is:
[tex]\[
\alpha \approx 53.7842^\circ
\][/tex]
Hence, when rounded to four decimal places, the angle [tex]\( \alpha = 53^\circ 47' 3'' \)[/tex] is approximately [tex]\( 53.7842^\circ \)[/tex].
