To find the decimal approximation for $\sec(39^\circ 18')$, follow these steps:

1. Convert the angle:

   • $39^\circ 18'$ means 39 degrees and 18 minutes.
   • There are 60 minutes in a degree, so $18'$ is $\frac{18}{60}$ degrees.

   Therefore: $39^\circ 18' = 39 + \frac{18}{60} = 39 + 0.3 = 39.3^\circ$

2. Calculate the secant:

   • The secant function is the reciprocal of the cosine function: $\sec(\theta) = \frac{1}{\cos(\theta)}$ Therefore, $\sec(39.3^\circ) = \frac{1}{\cos(39.3^\circ)}$.

3. Find the cosine:

   • Using a calculator, find the cosine of $39.3^\circ$. $\cos(39.3^\circ) \approx 0.777146$

4. Calculate the secant: $\sec(39.3^\circ) = \frac{1}{0.777146} \approx 1.286789$

Thus, the decimal approximation of $\sec(39^\circ 18')$ is approximately: $1.286789$

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If you'd like more details or have any questions, feel free to ask!

Further Exploration:

1. What is the value of $\sec(30^\circ)$?
2. How does secant relate to other trigonometric functions like sine and cosine?
3. How would the secant function behave at angles close to 90°?
4. Can you find the secant of an angle greater than 90°? What happens to it?
5. How does the unit circle help in understanding the secant function?

Tip: Secant functions are undefined for angles where the cosine is zero, such as 90° and 270°, because division by zero is not possible.