To convert the angle $\left(\frac{4}{7}\right)^c$ into degrees, minutes, and seconds, let's break down the process step by step:

Step 1: Express the Angle in Decimal Degrees

Given that $\left(\frac{4}{7}\right)^c$ means the fraction $\frac{4}{7}$ of 1 degree, first compute the decimal value of $\frac{4}{7}$:

$\frac{4}{7} \approx 0.57142857 \, \text{degrees}.$

Step 2: Convert Decimal Degrees to Minutes

There are 60 minutes in a degree. To find the minutes, multiply the decimal part by 60:

$0.57142857 \times 60 = 34.285714 \, \text{minutes}.$

Take the integer part (34 minutes), and move to the next step to convert the decimal part into seconds.

Step 3: Convert Decimal Minutes to Seconds

Now, take the decimal part of the minutes (0.285714) and convert it into seconds. Since there are 60 seconds in a minute, multiply by 60:

$0.285714 \times 60 = 17.14284 \, \text{seconds}.$

Thus, the seconds part is approximately 17 seconds.

Final Result:

The angle $\frac{4}{7}$ degrees is equivalent to:

$0^\circ 34' 17''$

This is the final answer in degrees, minutes, and seconds.

Would you like more details or have any questions about the process?

Here are five related questions to extend your understanding:

1. How do you convert a larger angle, say 5.25 degrees, into degrees, minutes, and seconds?
2. What is the relationship between degrees, minutes, and seconds in terms of conversion?
3. How do you convert from degrees, minutes, and seconds back to decimal degrees?
4. How would the method change for an angle in radians instead of degrees?
5. What practical applications do you see for this type of conversion in real life?

Tip: Always remember that there are 60 minutes in a degree and 60 seconds in a minute when working with angle conversions!