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.