To find \(\sin(\theta)\) given sides of a triangle, we can use the definition of sine. For a right triangle, if \(c\) is the length of the hypotenuse and \(b\) is one of the legs, then:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}. \]
In this case, we can consider:
- \(c = 17\) inches (the hypotenuse),
- \(b = 15\) inches (the adjacent side).
Since we need the opposite side for sine, we can find it using the Pythagorean theorem:
\[ c^2 = a^2 + b^2, \]
where \(a\) is the opposite side. Plugging in the values:
\[ 17^2 = a^2 + 15^2, \] \[ 289 = a^2 + 225, \] \[ a^2 = 289 - 225, \] \[ a^2 = 64, \] \[ a = 8 \text{ inches}. \]
Now that we have the opposite side, 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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