Use the Law of Sines to solve (if possible) for $c$ . Round your answers to two decimal places. $A = 110 ^ { \circ } , a = 125 , b = 100$
A) $c \approx 48.23$
B) $c \approx 68.23$
C) $c \approx 38.23$
D) $c \approx 58.23$
E) No solution

Question

Use the Law of Sines to solve (if possible) for $c$ . Round your answers to two decimal places. $A = 110 ^ { \circ } , a = 125 , b = 100$ 
A) $c \approx 48.23$
B) $c \approx 68.23$
C) $c \approx 38.23$
D) $c \approx 58.23$
E) No solution

Quizplus · Accepted Answer

The Law of Sines states that in a triangle with sides $a$, $b$, and $c$, and opposite angles $A$, $B$, and $C$, respectively, $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$In this case, we have $A = 110 ^ { \circ } , a = 125 , b = 100$, and we want to solve for $c$. Using the Law of Sines, we have$$\frac{c}{\sin C} = \frac{a}{\sin A}$$Solving for $c$, we get$$c = \frac{a \sin C}{\sin A}$$We don't know the value of $C$, but we can use the fact that the sum of the angles in a triangle is $180 ^ { \circ } $ to find it:$$A + B + C = 180 ^ { \circ }$$$$110 ^ { \circ } + B + C = 180 ^ { \circ }$$$$B + C = 70 ^ { \circ }$$Since $B$ and $C$ are opposite angles, we have$$C = 180 ^ { \circ } - B$$Substituting this into the equation for $c$, we get$$c = \frac{a \sin (180 ^ { \circ } - B)}{\sin A}$$$$c = \frac{a \sin B}{\sin A}$$Now we can plug in the values of $a$, $A$, and $b$ to get$$c = \frac{125 \sin 70 ^ { \circ }}{\sin 110 ^ { \circ } }$$$$c \approx 48.23$$Therefore, the correct answer is A.

Use the Law of Sines to Solve (If Possible) For c