Consider a triangle ABC for which ∠A = 120°, a = 31, b = 18. If such a triangle can not exist, then write NONE in each answer box. If there could be more than one such triangle, then enter dimensions for the one with the smallest value for side c. Finally, if there is a unique triangle ABC, then enter its dimensions.

∠B is degrees;

ZC is degrees;

c=

Given a triangle \( ABC \) with \( \angle A = 120^\circ \), \( a = 31 \), and \( b = 18 \), we need to determine the dimensions of the triangle, specifically \( \angle B \), \( \angle C \), and \( c \). 

We will use the Law of Cosines to find \( c \), and then the Law of Sines to find \( \angle B \) and \( \angle C \).

The Law of Cosines states:

\[ c^2 = a^2 + b^2 - 2ab \cos(\angle A) \]

Given:
\[ a = 31 \]
\[ b = 18 \]
\[ \angle A = 120^\circ \]

First, we calculate \( c \):

\[ c^2 = 31^2 + 18^2 - 2 \times 31 \times 18 \times \cos(120^\circ) \]

Since \( \cos(120^\circ) = -\frac{1}{2} \):

\[ c^2 = 31^2 + 18^2 + 2 \times 31 \times 18 \times \frac{1}{2} \]
\[ c^2 = 961 + 324 + 31 \times 18 \]
\[ c^2 = 961 + 324 + 279 \]
\[ c^2 = 1564 \]
\[ c = \sqrt{1564} \]
\[ c \approx 39.54 \]

Next, we use the Law of Sines to find \( \angle B \) and \( \angle C \):

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

First, find \( \sin A \):

\[ \sin(120^\circ) = \frac{\sqrt{3}}{2} \]

Now, solve for \( \sin B \):

\[ \frac{31}{\sin 120^\circ} = \frac{18}{\sin B} \]
\[ \frac{31}{\frac{\sqrt{3}}{2}} = \frac{18}{\sin B} \]
\[ \frac{31 \times 2}{\sqrt{3}} = \frac{18}{\sin B} \]
\[ \frac{62}{\sqrt{3}} = \frac{18}{\sin B} \]
\[ \sin B = \frac{18 \sqrt{3}}{62} \]
\[ \sin B = \frac{18 \sqrt{3}}{62} \approx 0.502 \]

Next, find \( \angle B \):

\[ \angle B \approx \sin^{-1}(0.502) \]
\[ \angle B \approx 30.1^\circ \]

Finally, calculate \( \angle C \):

\[ \angle C = 180^\circ - \angle A - \angle B \]
\[ \angle C = 180^\circ - 120^\circ - 30.1^\circ \]
\[ \angle C \approx 29.9^\circ \]

So, the dimensions of the unique triangle ABC are:

\[ \angle B \approx 30.1^\circ \]
\[ \angle C \approx 29.9^\circ \]
\[ c \approx 39.54 \]

Would you like more details or have any questions?

Here are 5 questions you might consider asking next:
1. How can you determine if a set of angles and sides can form a valid triangle?
2. Can you explain more about the Law of Cosines and its applications?
3. What are the other methods for solving triangles besides the Law of Sines and Cosines?
4. How do you handle ambiguous cases in triangle solutions?
5. What are some practical applications of these trigonometric principles?

Tip: When using trigonometric functions, make sure your calculator is set to the correct mode (degrees or radians) as required by the problem.

Explore how to solve triangle ABC with specific angles and sides using the Law of Cosines and Law of Sines. Understand the calculations for angles B and C, and side c in a triangle with angle A = 120°, side a = 31, and side b = 18. Suitable for high school students studying trigonometry and geometry.

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