4 A triangle has two angles that measures 110 degrees and 40 degrees. If the side opposite the 110 degroes angle measures 12 , find the length of the side opposite 40 degree angle. round your answer to nearest tenth.

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To find the length of the side opposite the 40-degree angle in the triangle, we can use the Law of Sines. The Law of Sines states that:

$$
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
$$

where $$a$$, $$b$$, and $$c$$ are the lengths of the sides opposite angles $$A$$, $$B$$, and $$C$$ respectively.

In this case:
- Let $$A = 110^\circ$$ (the angle opposite the side of length 12)
- Let $$B = 40^\circ$$ (the angle opposite the side we want to find, which we'll call $$b$$)
- The third angle $$C$$ can be found using the fact that the sum of angles in a triangle is $$180^\circ$$:

$$
C = 180^\circ - A - B = 180^\circ - 110^\circ - 40^\circ = 30^\circ
$$

Now we can set up the Law of Sines:

$$
\frac{12}{\sin(110^\circ)} = \frac{b}{\sin(40^\circ)}
$$

To find $$b$$, we can rearrange the equation:

$$
b = 12 \cdot \frac{\sin(40^\circ)}{\sin(110^\circ)}
$$

Now we need to calculate $$\sin(40^\circ)$$ and $$\sin(110^\circ)$$:

- $$\sin(40^\circ) \approx 0.6428$$
- $$\sin(110^\circ) \approx 0.9397$$

Now substituting these values into the equation:

$$
b = 12 \cdot \frac{0.6428}{0.9397}
$$

Calculating the fraction:

$$
\frac{0.6428}{0.9397} \approx 0.684
$$

Now multiply by 12:

$$
b \approx 12 \cdot 0.684 \approx 8.208
$$

Rounding to the nearest tenth, we find:

$$
b \approx 8.2
$$

Thus, the length of the side opposite the 40-degree angle is approximately **8.2**.
The length of the side opposite the 40-degree angle is approximately 8.2.

4 A triangle has two angles that measures 110 degrees and 40 degrees. If the side opposite the 110 degroes angle measures 12 , find the length of the side opposite 40 degree angle. round your answer to nearest tenth.

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