Solve side c when α=109∘, β=32∘ and side a=11

To solve for side c, we can use the Law of Sines. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Step 1: Write down the formula for the Law of Sines:
\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}

Step 2: Plug in the given information:
\frac{11}{\sin(109^\circ)} = \frac{b}{\sin(32^\circ)} = \frac{c}{\sin\gamma}

Step 3: Solve for side c:
\frac{11}{\sin(109^\circ)} = \frac{c}{\sin\gamma}

\sin\gamma = \frac{c}{\frac{11}{\sin(109^\circ)}}

\sin\gamma = \frac{c \cdot \sin(109^\circ)}{11}

Step 4: Plug in the known value of \beta (32 degrees) to find the value of \gamma :
\alpha+\beta+\gamma=180^{\circ}

109^{\circ}+32^{\circ}+\gamma=180^{\circ}

\gamma=180^{\circ}-32^{\circ}-109^{\circ}

\gamma=39^{\circ}

Step 5: Substitute the value of \sin\gamma and solve for side c:
\sin\gamma = \frac{c \cdot \sin(109^\circ)}{11}

\sin(39^{\circ})=\frac{c \cdot\sin(109^\circ)}{11}

Using a calculator, we find that \sin(39^{\circ})\approx0.6293 . So we can rewrite the equation as:

0.6293=\frac{c \cdot\sin(109^\circ)}{11}

Now, solve for c:
c\cdot\sin(109^{\circ})=11\cdot0.6293

c=\frac{11\cdot0.6293}{\sin(109^{\circ})}

c\approx\frac{6.9225}{0.9455}

c\approx7.3214

Answer: Side c ≈ 7.32