To find the value of \( c \) given that \( a = 6 \) inches and \( \theta = 45^\circ \), we can use trigonometric relationships. However, the nature of \( c \) must be specified; generally, it could refer to the length of a side in a right triangle where \( a \) is one of the legs, and \( \theta \) is the angle opposite that leg.
For example, if we are referring to a right triangle, we would use the sine, cosine, or tangent functions. Assuming \( c \) is the hypotenuse, we can use the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} \]
Substituting the known values:
\[ \sin(45^\circ) = \frac{6}{c} \]
Since \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \):
\[ \frac{\sqrt{2}}{2} = \frac{6}{c} \]
Now, rearranging for \( c \):
\[ c = \frac{6}{\frac{\sqrt{2}}{2}} = 6 \cdot \frac{2}{\sqrt{2}} = \frac{12}{\sqrt{2}} \]
To rationalize the denominator:
\[ c = \frac{12 \sqrt{2}}{2} = 6\sqrt{2} \]
Now we need to evaluate \( 6\sqrt{2} \):
\[ \sqrt{2} \approx 1.414 \] \[ c \approx 6 \cdot 1.414 \approx 8.484 \]
Rounding to the nearest hundredth, we get:
\[ c \approx 8.48 \text{ in.} \]
Therefore, the value of \( c \) is \( \boxed{8.48} \) inches.
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