Understanding the Given Equation
The equation provided is \( 7\pi = 2\alpha \). To find the value of \( \alpha \), we can rearrange it as follows:
- Divide both sides by 2:
\[
\alpha = \frac{7\pi}{2}
\]
This means that \( \alpha \) represents an angle in radians.
Finding \( x \)
To find \( x \), we note that \( \alpha \) is often related to angles in trigonometric functions. We'll take \( x = \frac{\alpha}{2} \):
- Substituting in the value of \( \alpha \):
\[
x = \frac{7\pi}{4}
\]
Calculating \( \sec x, \sec 2x, \sec 4x \)
Now, we need to find \( \sec x \), \( \sec 2x \), and \( \sec 4x \).
- Calculate \( \sec x \):
\[
\sec x = \sec\left(\frac{7\pi}{4}\right) = \frac{1}{\cos\left(\frac{7\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}
\]
- Calculate \( \sec 2x \):
\[
2x = \frac{7\pi}{2}
\]
\[
\sec 2x = \sec\left(\frac{7\pi}{2}\right) = \sec\left(\frac{3\pi}{2}\right) = \frac{1}{\cos\left(\frac{3\pi}{2}\right)} = \text{undefined}
\]
- Calculate \( \sec 4x \):
\[
4x = 14\pi
\]
\[
\sec 4x = \sec(14\pi) = \frac{1}{\cos(14\pi)} = \frac{1}{1} = 1
\]
Final Values
- \( \sec x = -\sqrt{2} \)
- \( \sec 2x = \text{undefined} \)
- \( \sec 4x = 1 \)