To find \(\sin \theta\), we can use the definition of sine in a right triangle. In this case, the sine of angle \(\theta\) is given by the ratio of the length of the opposite side (\(a\)) to the length of the hypotenuse (\(c\)).
From the problem, we have:
- \(c = 17\) inches (the length of the hypotenuse)
- \(b = 15\) inches (one of the other sides)
We can find the length of side \(a\) using the Pythagorean theorem: \[ c^2 = a^2 + b^2 \] Plugging in the values we have: \[ 17^2 = a^2 + 15^2 \] Calculating the squares: \[ 289 = a^2 + 225 \] Now, solve for \(a^2\): \[ a^2 = 289 - 225 \] \[ a^2 = 64 \] Taking the square root: \[ a = 8 \text{ inches} \]
Now that we have the lengths of all sides, we can find \(\sin \theta\): \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{8}{17} \]
So, the value of \(\sin \theta\) is \[ \frac{8}{17}. \]
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