If a distance of 75 yds is measured back from the edge of the canyon and two angles are measured , find the distance across the canyonAngle ACB = 50°Angle ABC=100°a=75 ydsWhat does C equal?

Answer:Length of side [tex]c[/tex]:[tex]c = \rm AB \approx 115\; yds[/tex].The distance across the canyon is approximately[tex]\rm 113\;yds[/tex].Step-by-step explanation:[tex]c[/tex] is the length of the side opposite to the angle [tex]\rm A\hat{C} B[/tex].[tex]a[/tex] is the length of the side opposite to the angle [tex]\rm B\hat{A}C[/tex].Apply the law of sine:[tex]\displaystyle \frac{c}{\sin{\rm A\hat{C}B} = \frac{a}{\sin{\rm B\hat{A}C}}[/tex].In other words, [tex]\displaystyle c = \sin{\rm B\hat{A}C} \cdot \frac{a}{\sin{\rm A\hat{C} B}}[/tex].However, the value of the angle [tex]\rm B\hat{A}C[/tex] isn't given. Don't panic. The three interior angles of a triangle shall add up to 180°. Two of the three angles are given. The value of the third angle is implied.[tex]\rm B\hat{A}C = 180\textdegree{} - A\hat{C}B - A\hat{B}C = 30\textdegree{}[/tex].Apply the law of sine to find [tex]c[/tex]:[tex]\displaystyle \begin{aligned}c &= \sin{\rm A\hat{C}B} \cdot \frac{a}{\sin{\rm B\hat{A}C}}\\ &= \sin{50\textdegree{}}\cdot \frac{\rm 75\; yds}{\sin{30\textdegree{}}}\\ &\rm = 114.907\; yds\end{aligned}[/tex].Refer to the diagram. The distance across the canyon will be[tex]\rm AB \cdot \sin{A\hat{B}C} = 114.907\times \sin{100\textdegree{}} \approx 113\;yds[/tex].