Questions: If (a=6 , textin). and (theta=45^circ), find the value of (c). Round your answer to the nearest hundredth.

Solution Steps

In a right triangle, the hypotenuse \( c \) can be found using the sine function: \[ \sin(\theta) = \frac{a}{c} \]

Given \( a = 6 \) inches and \( \theta = 45^\circ \), substitute these values into the equation: \[ \sin(45^\circ) = \frac{6}{c} \]

The sine of \( 45^\circ \) is \(\frac{\sqrt{2}}{2}\). Substitute this value: \[ \frac{\sqrt{2}}{2} = \frac{6}{c} \]

Rearrange the equation to solve for \( c \): \[ c = \frac{6 \times 2}{\sqrt{2}} = \frac{12}{\sqrt{2}} \]

Simplify \(\frac{12}{\sqrt{2}}\) by multiplying the numerator and the denominator by \(\sqrt{2}\): \[ c = \frac{12 \sqrt{2}}{2} = 6\sqrt{2} \]

Calculate \( 6\sqrt{2} \) to the nearest hundredth: \[ c \approx 6 \times 1.414 = 8.49 \]

Final Answer