Problem

Use the appropriate reciprocal identity to find the exact value of $\sin \theta$ for the given value of $\csc \theta$. Rationalize denominators when applicable. \[ \csc \theta=\frac{\sqrt{63}}{7} \] \[ \sin \theta= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution

Step 1 :We are given that \(\csc \theta = \frac{\sqrt{63}}{7}\)

Step 2 :We know that the reciprocal identity for sine and cosecant is \(\sin \theta = \frac{1}{\csc \theta}\)

Step 3 :So, to find the value of \(\sin \theta\), we need to take the reciprocal of the given value of \(\csc \theta\)

Step 4 :\(\sin \theta = \frac{1}{\frac{\sqrt{63}}{7}} = \frac{7}{\sqrt{63}}\)

Step 5 :We simplify the above expression to get \(\sin \theta = \frac{\sqrt{7}}{3}\)

Step 6 :Final Answer: \(\sin \theta = \boxed{\frac{\sqrt{7}}{3}}\)

From Solvely APP
Source: https://solvelyapp.com/problems/7562/

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