To find the remaining trigonometric functions for $\theta$, given that $\sec(\theta) = 7$, we can start by using the definition of the secant function and then use the Pythagorean identity to find $\sin(\theta)$.

1. Start with the given information: $\sec(\theta) = 7$ Since $\sec(\theta) = \frac{1}{\cos(\theta)}$, we have: $\cos(\theta) = \frac{1}{\sec(\theta)} = \frac{1}{7}$

2. Use the Pythagorean identity to find $\sin(\theta)$: The Pythagorean identity states: $\sin^2(\theta) + \cos^2(\theta) = 1$ Substitute $\cos(\theta) = \frac{1}{7}$: $\sin^2(\theta) + \left(\frac{1}{7}\right)^2 = 1$ $\sin^2(\theta) + \frac{1}{49} = 1$ Subtract $\frac{1}{49}$ from both sides: $\sin^2(\theta) = 1 - \frac{1}{49} = \frac{49 - 1}{49} = \frac{48}{49}$ Now, take the square root of both sides (since $\theta$ is acute, $\sin(\theta)$ is positive): $\sin(\theta) = \sqrt{\frac{48}{49}} = \frac{\sqrt{48}}{7} = \frac{4\sqrt{3}}{7}$

Thus, the exact value of $\sin(\theta)$ is: $\sin(\theta) = \frac{4\sqrt{3}}{7}$

Would you like me to find the other trigonometric functions? Let me know if you have questions.

Here are 5 related questions to expand understanding:

1. How do we find the value of $\tan(\theta)$ using $\sin(\theta)$ and $\cos(\theta)$?
2. How can we find $\cot(\theta)$ given $\tan(\theta)$?
3. What is the geometric interpretation of secant, and how does it relate to cosine?
4. How does knowing one trigonometric function help in finding all the others?
5. Why is $\sin(\theta)$ positive if $\theta$ is an acute angle?

Tip: Remember that each trigonometric function is interconnected; once you know one or two values, you can often use identities to find the rest!