The equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ can be used to find the length of $\overline{AB}$. What is the length of $\overline{AB}$?
The problem provides the equation \(\sin(25^{\circ}) = \frac{9}{c}\), which describes a relationship often found in right-angled triangles where the sine of an angle is equal to the ratio of the length of the opposite side to the length of the hypotenuse. In this context, the opposite side has a length of \(9\), and the hypotenuse is represented by the segment \(\overline{AB}\) with length \(c\). Our objective is to find the length \(c\).
To isolate the variable \(c\), we multiply both sides of the equation by \(c\) and then divide both sides by \(\sin(25^{\circ})\):
\[c \cdot \sin(25^{\circ}) = 9\]
\[c = \frac{9}{\sin(25^{\circ})}\]
This expression gives the exact length of \(\overline{AB}\).
Using the known value of \(\sin(25^{\circ}) \approx 0.422618\), we calculate the length:
\[c = \frac{9}{0.422618} \approx 21.2958\]
The length of the segment \(\overline{AB}\) is approximately \(21.30\).
In architectural design, if a support beam is inclined at an angle of \(25^{\circ}\) from the horizontal and has a vertical height clearance of \(9\) meters, you can use the sine ratio to calculate the required length of the beam. By setting up the equation \(\sin(25^{\circ}) = \frac{9}{\text{length}}\), you find that the beam must be approximately \(21.30\) meters long to ensure structural stability and the correct clearance. This application of trigonometry allows engineers to determine essential dimensions directly from given angles and height requirements.