Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle

thetaθ.

cosecant theta equals 10cscθ=10

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Part 1

sine theta equalssinθ=enter your response here

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

To find the exact value of \(\sin \theta\) given that \(\csc \theta = 10\), we start by using the reciprocal identity that relates cosecant and sine:

\[
\csc \theta = \frac{1}{\sin \theta}
\]

Since \(\csc \theta = 10\), we have:

\[
10 = \frac{1}{\sin \theta}
\]

To solve for \(\sin \theta\), take the reciprocal of both sides:

\[
\sin \theta = \frac{1}{10}
\]

So, the exact value of \(\sin \theta\) is:

\[
\sin \theta = \frac{1}{10}
\]

Would you like to proceed with finding the remaining trigonometric functions or need further explanation on any step?

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Here are some related questions to deepen your understanding:

1. How do you find \(\cos \theta\) given \(\sin \theta\) using the Pythagorean identity?
2. What is the relationship between \(\sec \theta\) and \(\cos \theta\)?
3. How do we find \(\tan \theta\) using the values of \(\sin \theta\) and \(\cos \theta\)?
4. Can you explain why the reciprocal identities are helpful in trigonometry?
5. How would you find \(\cot \theta\) once \(\tan \theta\) is known?

**Tip:** Remember, knowing just one trigonometric function's value often allows you to determine all six, especially with identities and the Pythagorean theorem.

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle θ. Given that csc θ = 10, find sin θ. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Learn how to find the exact values of all trigonometric functions of an acute angle θ given that csc θ = 10. This solution covers reciprocal identities, the Pythagorean identity, and detailed steps to calculate sin θ, cos θ, tan θ, sec θ, and cot θ. Suitable for high school students in Grades 10-12.

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