Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle theta.
secant theta equals 7
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Part 1
sine theta equals

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​(Simplify your​ answer, including any radicals. Use integers or fractions for any numbers in the​ expression.)

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!

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle theta. secant theta equals 7

Discover how to determine the exact values of the five remaining trigonometric functions for an acute angle θ when sec(θ) = 7. This solution uses trigonometric identities, including the Pythagorean Identity, to calculate values for sin(θ), cos(θ), and tan(θ), providing a clear breakdown suitable for high school students.

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