To solve for \( c \) given that \( a = 6 \) inches and \( \theta = 45^\circ \), we can use trigonometric relationships. Assuming you are referring to a right triangle where \( a \) is one leg, \( c \) is the hypotenuse, and \( \theta \) is the angle opposite leg \( a \), we can use the sine function:
\[ \sin(\theta) = \frac{a}{c} \]
Plugging in the values, we have:
\[ \sin(45^\circ) = \frac{6}{c} \]
Since \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \), the equation becomes:
\[ \frac{\sqrt{2}}{2} = \frac{6}{c} \]
Now, we can solve for \( c \):
- Cross-multiply:
\[ \sqrt{2} \cdot c = 12 \]
- Divide both sides by \( \sqrt{2} \):
\[ c = \frac{12}{\sqrt{2}} \]
- To rationalize the denominator:
\[ c = \frac{12 \sqrt{2}}{2} = 6\sqrt{2} \]
- Finally, we'll calculate the value:
Using the approximation \( \sqrt{2} \approx 1.414 \):
\[ c \approx 6 \cdot 1.414 \approx 8.484 \]
Rounding to the nearest hundredth, we get:
\[ \boxed{8.48} \]
This is the value of \( c \).
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