Decide on the procedure to solve the triangle given the following information, then solve the triangle. $$ \mathrm{a}=51.41, \mathrm{~b}=37.11, \mathrm{c}=66.53 $$ Which of the following procedures should be used to solve the triangle? A. Use the law of cosines to find the largest angle. Use the law of sines to find either remaining angle. Use the angle sum formula to find the remaining angle. B. Use the angle sum formula to find the remaining angle and the law of sines to find the remaining sides. C. Use the law of cosines to find the third side and the law of sines to find the smaller of the two remaining angles. Use the angle sum formula find the remaining angle. D. Use the law of sines to find an angle. Then use the angle sum formula to find the remaining angle. Use the law of sines to find the remaining side.

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To solve the triangle given the sides $$ a = 51.41 $$, $$ b = 37.11 $$, and $$ c = 66.53 $$, we can determine the best procedure to use.

Since we have all three sides of the triangle, we can use the Law of Cosines to find one of the angles. The Law of Cosines is particularly useful when we have the lengths of all three sides. After finding one angle, we can then use the Law of Sines to find one of the remaining angles, and finally use the angle sum property to find the last angle.

Thus, the best procedure to use is:

**A. Use the law of cosines to find the largest angle. Use the law of sines to find either remaining angle. Use the angle sum formula to find the remaining angle.**

Now, let's solve the triangle step by step.

### Step 1: Find the largest angle using the Law of Cosines

The largest angle is opposite the longest side, which is side $$ c $$. We can use the Law of Cosines:

$$
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
$$

Rearranging to find $$ C $$:

$$
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
$$

Substituting the values:

$$
\cos(C) = \frac{51.41^2 + 37.11^2 - 66.53^2}{2 \cdot 51.41 \cdot 37.11}
$$

Calculating the squares:

$$
51.41^2 \approx 2643.5681
$$
$$
37.11^2 \approx 1373.6321
$$
$$
66.53^2 \approx 4425.2609
$$

Now substituting these values into the cosine formula:

$$
\cos(C) = \frac{2643.5681 + 1373.6321 - 4425.2609}{2 \cdot 51.41 \cdot 37.11}
$$
$$
\cos(C) = \frac{-408.0607}{2 \cdot 51.41 \cdot 37.11}
$$
$$
\cos(C) = \frac{-408.0607}{3815.8821} \approx -0.1068
$$

Now, we find $$ C $$:

$$
C \approx \cos^{-1}(-0.1068) \approx 96.13^\circ
$$

### Step 2: Use the Law of Sines to find another angle

Now we can use the Law of Sines to find angle $$ A $$:

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

Rearranging gives:

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

Substituting the known values:

$$
\sin(A) = \frac{51.41 \cdot \sin(96.13^\circ)}{66.53}
$$

Calculating $$ \sin(96.13^\circ) $$:

$$
\sin(96.13^\circ) \approx 0.9962
$$

Now substituting:

$$
\sin(A) = \frac{51.41 \cdot 0.9962}{66.53} \approx \frac{51.174}{66.53} \approx 0.769
$$

Now, we find $$ A $$:

$$
A \approx \sin^{-1}(0.769) \approx 50.48^\circ
$$

### Step 3: Find the remaining angle $$ B $$

Using the angle sum property:

$$
B = 180^\circ - A - C
$$
$$
B = 180^\circ - 50.48^\circ - 96.13^\circ \approx 33.39^\circ
$$

### Summary of the triangle

- $$ A \approx 50.48^\circ $$
- $$ B \approx 33.39^\circ $$
- $$ C \approx 96.13^\circ $$

Thus, the triangle is solved with the angles approximately $$ A \approx 50.48^\circ $$, $$ B \approx 33.39^\circ $$, and $$ C \approx 96.13^\circ $$.
To solve the triangle with sides $$ a = 51.41 $$, $$ b = 37.11 $$, and $$ c = 66.53 $$, use the Law of Cosines to find the largest angle $$ C $$, then use the Law of Sines to find angle $$ A $$, and finally use the angle sum formula to find angle $$ B $$. The angles are approximately $$ A \approx 50.48^\circ $$, $$ B \approx 33.39^\circ $$, and $$ C \approx 96.13^\circ $$.

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