Use the Law of Sines to solve for the remaining side(s) and angle(s) of the triangle if $α = 14^{\circ}, β = 19^{\circ}, a = 9$ . Round to two decimal places.

As in the text, $(α, a), (β, b)$ and $(γ, c)$ are angle-side opposite pairs. If no such triangle exists, enter DNE in each answer box.
$\begin{array}{l} γ = ◻ degrees \\ b = ◻ \\ c = ◻ \end{array}$

Answer

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Answer

Final Answer: $γ = 147.0$ degrees, $b = 12.11$ , and $c = 20.26$ .

Steps

Step 1 ：Given the angles $α = 14^{\circ}$ and $β = 19^{\circ}$ , and the side $a = 9$ .

Step 2 ：First, convert the angles from degrees to radians because the trigonometric functions work with radians.

Step 3 ：Calculate $γ$ using the formula $γ = π - α - β$ because the sum of angles in a triangle is $π$ radians.

Step 4 ：Calculate $b$ and $c$ using the Law of Sines: $b = a \cdot \frac{\sin (β)}{\sin (α)}$ and $c = a \cdot \frac{\sin (γ)}{\sin (α)}$ .

Step 5 ：Convert $γ$ back to degrees.

Step 6 ：Round $γ$ , $b$ , and $c$ to two decimal places.

Step 7 ：Final Answer: $γ = 147.0$ degrees, $b = 12.11$ , and $c = 20.26$ .