One of the angles in a right triangle is \( \theta=74.0^{\circ} \), the side adjacent to the angle has a length 31.0 m . Find the length of the hypotenuse of the triangle: Find the length of the side opposite to \( \theta: \) Express your answers with the correct number of sifnificant figures.

To find the length of the hypotenuse \( c \) in a right triangle where \( \theta = 74.0^{\circ} \) and the length of the adjacent side \( a = 31.0 \, \text{m} \), we can use the cosine function: \[ \cos(74.0^{\circ}) = \frac{a}{c} \] Rearranging gives: \[ c = \frac{a}{\cos(74.0^{\circ})} = \frac{31.0 \, \text{m}}{\cos(74.0^{\circ})} \approx \frac{31.0}{0.276} \approx 112.9 \, \text{m} \] Thus, the length of the hypotenuse is approximately \( 112.9 \, \text{m} \) when rounded to three significant figures. Next, to find the length of the side opposite \( \theta \) (let's call it \( b \)), we can use the sine function: \[ \sin(74.0^{\circ}) = \frac{b}{c} \] We already calculated \( c \). Rearranging gives: \[ b = c \cdot \sin(74.0^{\circ}) \approx 112.9 \, \text{m} \cdot \sin(74.0^{\circ}) \approx 112.9 \cdot 0.961 \approx 108.7 \, \text{m} \] So, the length of the side opposite \( \theta \) is approximately \( 108.7 \, \text{m} \) when rounded to three significant figures.